It's not mandatory that you get an exhaust along with an intake, but to get the best performance, it's helpful.

As previously stated (But I'll say much shorter ) the more air you car can take in and the faster it can be pushed out the better.
Of course if you're getting more air in, more fuel has to be mixed thus, creating more power.
For example

Imagine your stock intake is one of those little stirrer straws you use to stir your coffee. Imagine drinking from that straw, Pretty restrictive right.

Now Imagine an aftermarket intake is a Mcdonalds straw. Drinking through that straw is much easier

The same with an exhaust system (header(s)..pipes..muffler)
Try blowing through that stirrer (which is your stock exhaust) and then blow through a Mcdonalds straw (aftermarket exhaust)