I would organize this into a class with methods first. This breaks down each piece of logic and makes it easier to read.

1. Start a "play" loop
2. Inside the "play" loop, have a "guess" loop
3. Store the "continue" state in a known constant
4. Only check if they are correct after they made the max number of guesses.
5. Seed the Random object with the current system time

Also, I added a DEBUG flag to display the answer.

Update: The program is ignoring or not even asking for your "yes" input, because you were previously asking it for integers. You will need to clear/flush/reset the scanner's buffer by calling nextLine. Switching from integer to string leaves some data in the buffer. So your program will use the left-over character data in the buffer before it prompts you for your input.

import java.util.Scanner;
import java.util.Random;

public class HiLo implements Runnable {
    private static boolean DEBUG = true; // Disable to not show the answer...
    private static final int MAX_NUMBER = 100;
    private static final int MAX_GUESSES = 5;
    private static final String YES = "YES";

    private final Random rand;

    public HiLo() {
        this.rand = new Random(System.currentTimeMillis());
    }

    public static void main(String[] args) {
        new Thread(new HiLo()).start();
    }

    @Override
    public void run() {
        Scanner scan = new Scanner(System.in);
        String cont = "";
        do {
            play(scan);
            System.out.print("\nWould you like to play again (yes/no)? ");
            cont = scan.nextLine();
            System.out.println();
        } while (cont.equalsIgnoreCase(YES));
        scan.close();
    }

    public void play(Scanner scan) {
        boolean correct = false;
        int numberOfGuesses = 0;
        int answer = rand.nextInt(MAX_NUMBER) + 1;

        if (DEBUG) System.out.printf("The answer is: %d%n", answer);

        System.out.printf("Guess a number between 1 and %d%n", MAX_NUMBER);

        while (numberOfGuesses < MAX_GUESSES && !correct) {
            numberOfGuesses++;
            correct = makeGuess(scan, answer, numberOfGuesses);
        }

        if (numberOfGuesses <= MAX_GUESSES && correct) {
            System.out.printf("Correct! The number was %d. It took you %d guesses.%n", answer, numberOfGuesses);
        } else {
            System.out.printf("Sorry, but you did not guess: %d", answer);
        }
        scan.nextLine(); // Reset...
    }

    public boolean makeGuess(Scanner scan, int answer, int currentGuesses) {
        System.out.print("Guess: ");
        int guess = scan.nextInt();

        if (guess == answer) {
            return true;
        }

        if (currentGuesses < MAX_GUESSES) {
            if (guess > answer) {
                System.out.print("Too High, ");
            } else {
                System.out.print("Too Low, ");
            }
        }

        return false;
    }
}

Sample Output

The answer is: 64
Guess a number between 1 and 100
Guess: 50
Too Low, Guess: 75
Too High, Guess: 60
Too Low, Guess: 65
Too High, Guess: 64
Correct! The number was 64. It took you 5 guesses.

Would you like to play again (yes/no)? yes

The answer is: 88
Guess a number between 1 and 100
Guess: 50
Too Low, Guess: 90
Too High, Guess: 80
Too Low, Guess: 85
Too Low, Guess: 87
Sorry, but you did not guess: 88

Would you like to play again (yes/no)? no

A possible solution is to place the game in another while loop(that envelops that of the gameCount) and within the while loop create a switch statement for a variable lets say gameContinue. At the end of the game the user would receive a prompt saying whether to continue or not. Based on their answer you change the value of the variable gameContinue and there you go.

You have loaded all your code into the main() method. That is generally not good. You should aim to keep main() relatively small and have the workings hidden in other methods that you call from main(). Use main() more for flow control than for actually playing the game. Something like:

import java.util.Random;
import java.util.Scanner;

public class GuessingGameTest {

  // Shared class utilities.
  Random myRand = new Random();
  Scanner inScan = new Scanner(System.in);

  public static void main(String[] args) {

    // Introduce the game.
    System.out.println("Hello and welcome to my number guessing game.");

    // Play the game.
    boolean playing = true;
    while (playing) {

      // Play one game.
      playGame();

      // Another game?
      char reply = 'X';
      while (reply != '1' && reply != '2')
        System.out.println("Do you want to start again?");
        System.out.println("Type 1 to start again, type 2 to quit.");
        reply = inScan.nextChar();
      }

      if (reply == '2') {
        playing = false;  // Exit game.
      }
    }

    System.out.println("Thank you for playing.  Goodbye.");

  }

  static void playGame() {
    System.out.println("Pick a number: ");
    // Much code here.
  }

} // end class GuessingGameTest.

I have not compiled or tested this code, so test it thoroughly before using it.

Your use of exit() is not good practice either. Let the code finish naturally, and reserve exit() for exceptional situations, not for a normal termination.

You need to exit the while when correct.

To do so, after the line:

System.out.println("Correct!");

Add another line like this:

break;

Build a while (or do-while, which would be even a better choice) around it. Ask if it should be played again, if yes the condition of the while stays true, otherwise change it. Of course you have to initialize numSticks, numToTake and score to 0 again.

package test;

import java.io.BufferedReader;
import java.io.InputStreamReader;
import java.util.Scanner;

public class Sticks3 {

    public static void main(String[] args) throws java.lang.Exception {
        BufferedReader br = new BufferedReader(new InputStreamReader(System.in));

        int numSticks = 21;
        System.out.println("Would You Like to go first? (Yes/No)");
        Scanner input = new Scanner(System.in);
        String goFirst = input.nextLine();
        Scanner take = new Scanner(System.in);
        int numToTake = 0;
        int score = 0;

        boolean playAgain = true;
        Scanner askPlayAgainScanner = new Scanner (System.in);

        while (playAgain == true) {

            numSticks = 21;
            numToTake = 0;
            score = 0;

            while (numSticks > 0) {

                if (goFirst.equals("Yes") || goFirst.equals("yes")) {
                    System.out.println("There are " + numSticks + " sticks ");
                    System.out.println("How many sticks do you want to take (1 or 2)");
                    numToTake = take.nextInt();

                    if (numToTake > 2) {
                        numToTake = 2;

                    }

                    else if (numToTake < 1) {
                        numToTake = 1;
                    }
                    numSticks = numSticks - numToTake;

                    if (numSticks <= 0) {
                        System.out.println("You lose");
                        System.out.println("Your score is " + score);
                    } else {

                        if ((numSticks - 2) % 3 == 0 || numSticks - 2 == 0) {
                            numToTake = 1;
                        } else {
                            numToTake = 2;
                        }
                        System.out.println("Computer takes " + numToTake + " sticks ");
                        numSticks = numSticks - numToTake;

                        if (numSticks <= 0) {
                            System.out.println(" You win ");
                            score++;
                            System.out.println("Your score is " + score);
                        }
                    }

                } else {
                    if ((numSticks - 2) % 3 == 0 || numSticks - 2 == 0) {
                        numToTake = 1;
                    } else {
                        numToTake = 2;
                    }
                    System.out.println("Computer takes" + numToTake + " sticks ");
                    numSticks = numSticks - numToTake;

                    if (numSticks <= 0) {
                        System.out.println("You win");
                        score++;
                        System.out.println("Your score is " + score);
                    } else {
                        System.out.println("There are " + numSticks + " sticks ");
                        System.out.println("How many sticks do you want to take (1 or 2)");
                        numToTake = take.nextInt();

                        if (numToTake > 2) {
                            numToTake = 2;
                        } else if (numToTake < 1) {
                            numToTake = 1;
                        }
                        numSticks = numSticks - numToTake;

                        if (numSticks <= 0) {
                            System.out.println("You win");
                            score++;
                            System.out.println("Your score is " + score);

                        }



                    }

                }
            }
            System.out.print("Do you want to play again? true for yes, false for no: ");
            boolean askPlayAgain = askPlayAgainScanner.nextBoolean();
        }
    }
}

Try this. Basically, if the user responds with "yes" , we will call the function again.

    if (tries > 9) {
            System.out.println("You should be able to do better!"
                    + " You have hit your ten guess limit.  The number" + " was: " + answer);
            System.out.println("Do you want to play again? (yes/no): "); // modified line
            if("yes".equalsIgnoreCase(input.next())){ // newly added if block
                answer = generateRandomNumber();
                tries=0;
                i=0;
                win = false;
                doGuess();
            }
            return;
        }