I am studying about maximum likelihood estimation and I read that the likelihood function is the product of the probabilities of each variable. Why is it the product? Why not the sum? I have been trying to search on Google but I can't find any meaningful answers.

https://en.wikipedia.org/wiki/Maximum_likelihood

This is a very basic question, and instead of using formal language and mathematical notation, I will try to answer it at a level at which everybody who can understand the question can also understand the answer. 

Imagine that we have a race of cats. They have a 75% probability of being born white, and 25% probability of being born grey, no other colors. Also, they have a 50% probability of having green eyes and 50% probability of having blue eyes, and coat color and eye color are independent. 

Now let us look at a litter of eight kittens: 



You will see that 1 out of 4, or 25%, are grey. Also, 1 out of 2, or 50% have blue eyes. Now the question is, 


  how many kittens have grey fur and blue eyes? 


You can count them, the answer is one. That is, $\frac{1}{4} \times \frac{1}{2} = \frac{1}{8}$, or 12.5% of 8 kittens. 

Why does it happen? Because any cat has a 1 in 4 probability to be grey. So, pick four cats, and you can expect one of them to be grey. But if you only pick four cats out of many (and get the expected value of 1 grey cat), the one which is grey has a 1 in 2 probability to have blue eyes. This means, of the total of cats you pick, you first multiply the total by 25% to get the grey cats, and then you multiply the selected 25% of all cats by 50% to get those of them which have blue eyes. This gives you the probability of getting blue-eyed grey cats. 

Summing them up would give you $\frac{1}{4} + \frac{1}{2}$, which makes $\frac{3}{4}$ or 6 out of 8. In our picture, it corresponds to summing up the cats which have blue eyes with the cats which have grey fur - and counting the one grey blue-eyed kitten twice! Such a calculation can have its place, but it is rather unusual in probability calculations, and it is certainly not the one you are asking about. 
