Equation for impulse train as sum of complex exponentials
The summation on the left side of your equation represents a time domain discrete time periodic signal $x[n]$ whose period is N.
$$ x[n] = \sum_{k=-\infty}^{\infty} \delta[n-kN] $$  

And the summation on the right side of your equation is just the same (periodic) signal $x[n]$ represented via its Discrete Fourier Series DFS synthesis form which is an alternative representation for periodic signals 
$$ x[n] = \sum_{k=0}^{N-1} a_k e^{j \frac{2 \pi}{N} k n} $$ 

where the DFS coefficients $a_k$ are all $1/N$ found via DFS analysis equation:
$$ a_k = \frac{1}{N} \sum_{n=0}^{N-1} x[n] e^{-j \frac{2 \pi}{N} n k} $$ 

I hope you can see why all $a_k$ are $1/N$ (it can be easily seen from the properties of impulse inside the summation...)