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Do an inverse transform, and plug in the expression for b_m in terms of a_n. For it to be "inverse", you should get
a_k = sum_n a_n delta_{n,k}, where delta_{n,k}=1 if n=k and 0 otherwise (which in turn would mean the trivial a_k=a_k.)
What you'll actually get instead of delta_{n,k} would be a sum over exponentials. If n is not equal to k, this sum is zero. If n=k, each term =1 and there are N terms, so you need to divide by N to get delta_{k,k}=1.
Now, there are two possible definitions. One is to do a direct transform without any prefactor, but then you would need to use 1/N prefactor in the inverse transform. Some people do that, so you might have seen that in the past. It's much nicer though to use a 1/sqrt(N) factor in both direct and inverse transforms (so that you use the same formula each time). Multiplied together they would give you the 1/N that you need. That's the formula in your book.
