Let's say you have 5 people working on the same set of code. Each of of those 5 people are making updates to the same set of files. Now you may click the build button and you know that you're code works, but what about when you integrate it with everyone else. The only you'll know is that if you get everyone else's and try. This is easy every once in a while, but it quickly becomes tiresome to do this over and over again.
With a build server that does it automatically, it checks if the code compiles for everyone all the time. Everyone always knows if the something is wrong with the build, and what the problem is, and no one has to do any work to figure it out. Small things add up, it may take a couple of minutes to pull down the latest code and try and compile it, but doing that 10-20 times a day quickly becomes a waste of time, especially if you have multiple people doing it. Sure you can get by without it, but it is so much easier to let an automated process do the same thing over and over again, then having a real person do it.
Here's another cool thing too. Our process is setup to test all the sql scripts as well. Can't do that with pressing the build button. It reloads snapshots of all the databases it needs to apply patches to and runs them to make sure that they all work, and run in the order they are supposed to. The build server is also smart enough to run all the unit tests/automation tests and return the results. Making sure it can compile is fine, but with an automation server, it can handle many many steps automatically that would take a person maybe an hour to do.
Taking this a step further, if you have an automated deployment process along with the build server, the deployment is automatic. Anyone who can press a button to run the process and deploy can move code to qa or production. This means that a programmer doesn't have to spend time doing it manually, which is error prone. When we didn't have the process, it was always a crap shoot as to whether or not everything would be installed correctly, and generally it was a network admin or a programmer who had to do it, because they had to know how to configure IIS and move the files. Now even our most junior qa person can refresh the server, because all they need to know is what button to push.