Chernoff bounds allow us to get the concentration of sum of independent binary random variables. (We can relax the independence to negative correlation/association etc). The standard proof goes by considering the idea of Bernstein to use moment generating function which allows us to decouple the variables. That is using the fact that exponentials behave nicely under addition.
Applying Markov’s inequality and optimizing the parameter 
The following paper gives a constructive proof of the bound. Here is how the proof goes. If the probability that the sum of the variables is much larger than the expectation is not too small, then we can by sampling a subset of the variables, we get that the probability that the subset of variables are all 
Note that the proof is essentially by counting!
A Neighbourhood Of Infinity 