I think the best way is to step through the code manually and write down the line number of the executing code and how the variables change.

Like

L01:    int k,i,j,s;                // k=?, i=?, j=?, s=?
L02:    scanf("%d", &k);            // k=18, i=?, j=?, s=?
L03:    for(i=1;                    // k=18, i=1, j=?, s=?
L03:             i<=k;              // TRUE
L04:        s=0;                    // k=18, i=1, j=?, s=0
L05:        for(j=1;                // k=18, i=1, j=1, s=0
L05:                 j<i;           // FALSE
L03:                      i++)      // k=18, i=2, j=1, s=0
L03:             i<=k;              // TRUE
L04:        s=0;                    // k=18, i=2, j=1, s=0
L05:        for(j=1;                // k=18, i=2, j=1, s=0
L05:                 j<i;           // TRUE
L06:            if(i%j==0)          // TRUE
L07:                s=s+j:          // k=18, i=2, j=1, s=1
L05:                      j++)      // k=18, i=2, j=2, s=0
L05:                  j<i;          // FALSE
and so on ....

I takes quite some time but you should soon see the pattern and there by understand how for-loops works.

Another thing that might help you understanding for-loops are to realize that

for(i=0; i<N; i++)
{
    code...
}

is equivalent to

i=0;
while (i<N)
{
    code...

     i++;
}

BTW:

Always check the return value from scanf - like:

if (scanf("%d", &k) != 1)
{
    printf("Input error! Program terminates.\n");
    exit(1);
}

in what situation would you need it

generally you would require a nested for loop when

you have two (or more) dimensions to work with

   for e.g :  working with images, x and y dimensions

              working with matrix

explanation of your code:

first you select a row to work with,

then in that selected row, you traverse all the columns

then you move on the second row

the outer loop takes care of moving to the second row, when all the columns of the first row have been processed/read

the inner for loop traverses the columns in the row selected by the outer loop

logic behind it:

the outer loop waits for the inner loop to finish, then moves on to its own next iteration.

i.e. for every iteration of the outer loop, the inner loop will cover its whole iterations.

as in your case, for every iteration of your outer loop the inner loop runs for 4 times (number of columns in each row)

and this process happens for all the rows, i.e. three times (number of rows)