I have added some comments to your solution:

def commonChild(s1, s2):
        # Allocate memory of size ([s1] + 1) * ([s2] + 1)
        # to contain the max child length starting with empty string.
        dp = [[0] * (len(s2) + 1) for _ in range(len(s1) + 1)]
        # Compare every pair of characters in both strings.
        for i, c1 in enumerate(s1):
                for j, c2 in enumerate(s2):
                        # If matching, add up to the score of the previous pair.
                        if c1 == c2:
                                dp[i + 1][j + 1] = dp[i][j] + 1
                        # Otherwise, paste the max score for two previous 
                        # reachable pairs.
                        else:
                                dp[i + 1][j + 1] = max(dp[i + 1][j], dp[i][j + 1])

# As the max score is being brought further through iterations,
# return the score for the last pair.
return dp[-1][-1]