The function works as follows:

# Import the cv2 library
import cv2
# Read the image you want connected components of
src = cv2.imread('/directorypath/image.bmp')
# Threshold it so it becomes binary
ret, thresh = cv2.threshold(src,0,255,cv2.THRESH_BINARY+cv2.THRESH_OTSU)
# You need to choose 4 or 8 for connectivity type
connectivity = 4  
# Perform the operation
output = cv2.connectedComponentsWithStats(thresh, connectivity, cv2.CV_32S)
# Get the results
# The first cell is the number of labels
num_labels = output[0]
# The second cell is the label matrix
labels = output[1]
# The third cell is the stat matrix
stats = output[2]
# The fourth cell is the centroid matrix
centroids = output[3]

Labels is a matrix the size of the input image where each element has a value equal to its label.

Stats is a matrix of the stats that the function calculates. It has a length equal to the number of labels and a width equal to the number of stats. It can be used with the OpenCV documentation for it:

Statistics output for each label, including the background label, see below for available statistics. Statistics are accessed via stats[label, COLUMN] where available columns are defined below.

• cv2.CC_STAT_LEFT The leftmost (x) coordinate which is the inclusive start of the bounding box in the horizontal direction.
• cv2.CC_STAT_TOP The topmost (y) coordinate which is the inclusive start of the bounding box in the vertical direction.
• cv2.CC_STAT_WIDTH The horizontal size of the bounding box
• cv2.CC_STAT_HEIGHT The vertical size of the bounding box
• cv2.CC_STAT_AREA The total area (in pixels) of the connected component

Centroids is a matrix with the x and y locations of each centroid. The row in this matrix corresponds to the label number.

The OpenCV 3.0 docs for connectedComponents() don't mention Python but it actually is implemented. See for e.g. this SO question. On OpenCV 3.4.0 and above, the docs do include the Python signatures, as can be seen on the current master docs.

The function call is simple: num_labels, labels_im = cv2.connectedComponents(img) and you can specify a parameter connectivity to check for 4- or 8-way (default) connectivity. The difference is that 4-way connectivity just checks the top, bottom, left, and right pixels and sees if they connect; 8-way checks if any of the eight neighboring pixels connect. If you have diagonal connections (like you do here) you should specify connectivity=8. Note that it just numbers each component and gives them increasing integer labels starting at 0. So all the zeros are connected, all the ones are connected, etc. If you want to visualize them, you can map those numbers to specific colors. I like to map them to different hues, combine them into an HSV image, and then convert to BGR to display. Here's an example with your image:

import cv2
import numpy as np

img = cv2.imread('eGaIy.jpg', 0)
img = cv2.threshold(img, 127, 255, cv2.THRESH_BINARY)[1]  # ensure binary
num_labels, labels_im = cv2.connectedComponents(img)

def imshow_components(labels):
    # Map component labels to hue val
    label_hue = np.uint8(179*labels/np.max(labels))
    blank_ch = 255*np.ones_like(label_hue)
    labeled_img = cv2.merge([label_hue, blank_ch, blank_ch])

    # cvt to BGR for display
    labeled_img = cv2.cvtColor(labeled_img, cv2.COLOR_HSV2BGR)

    # set bg label to black
    labeled_img[label_hue==0] = 0

    cv2.imshow('labeled.png', labeled_img)
    cv2.waitKey()

imshow_components(labels_im)

image = cv2.imread('image.png', cv2.IMREAD_UNCHANGED);
gray = cv2.cvtColor(image, cv2.COLOR_BGR2GRAY)
binary = cv2.threshold(gray, 0, 255, cv2.THRESH_BINARY | cv2.THRESH_OTSU)[1]

# getting mask with connectComponents
ret, labels = cv2.connectedComponents(binary)
for label in range(1,ret):
    mask = np.array(labels, dtype=np.uint8)
    mask[labels == label] = 255
    cv2.imshow('component',mask)
    cv2.waitKey(0)

# getting ROIs with findContours
contours = cv2.findContours(binary, cv2.RETR_EXTERNAL, cv2.CHAIN_APPROX_SIMPLE)[1]
for cnt in contours:
    (x,y,w,h) = cv2.boundingRect(cnt)
    ROI = image[y:y+h,x:x+w]
    cv2.imshow('ROI', ROI)
    cv2.waitKey(0)

cv2.destroyAllWindows()

In python, you should avoid deep loop. Prefer to use numpy other than python-loop.

Imporved:

##################################################
ts = time.time()
num = labels.max()

N = 50

## If the count of pixels less than a threshold, then set pixels to `0`.
for i in range(1, num+1):
    pts =  np.where(labels == i)
    if len(pts[0]) < N:
        labels[pts] = 0

print("Time passed: {:.3f} ms".format(1000*(time.time()-ts)))
# Time passed: 4.607 ms

##################################################

Result:

The whole code:

#!/usr/bin/python3
# 2018.01.17 22:36:20 CST
import cv2
import numpy as np
import time

img = cv2.imread('test.jpg', 0)
img = cv2.threshold(img, 127, 255, cv2.THRESH_BINARY)[1]  # ensure binary
retval, labels = cv2.connectedComponents(img)

##################################################
ts = time.time()
num = labels.max()

N = 50
for i in range(1, num+1):
    pts =  np.where(labels == i)
    if len(pts[0]) < N:
        labels[pts] = 0

print("Time passed: {:.3f} ms".format(1000*(time.time()-ts)))
# Time passed: 4.607 ms

##################################################

# Map component labels to hue val
label_hue = np.uint8(179*labels/np.max(labels))
blank_ch = 255*np.ones_like(label_hue)
labeled_img = cv2.merge([label_hue, blank_ch, blank_ch])

# cvt to BGR for display
labeled_img = cv2.cvtColor(labeled_img, cv2.COLOR_HSV2BGR)

# set bg label to black
labeled_img[label_hue==0] = 0

cv2.imshow('labeled.png', labeled_img)
cv2.imwrite("labeled.png", labeled_img)
cv2.waitKey()