What the check boxes do is create what is called a Service for that program. A simple way to think of a service is once you check that box, it will automatically start next time the computer boots up (and will continue to do so until the service is removed, mainly by un-checking the box again) So if you check apache, it will automatically start apache on boot up, instead of requiring you to click (Start)
This is obviously nice to have if you are working on a server remotely or if windows updates kick in at 2 am and your web server reboots, kicking apache and mysql offline. On the other hand, all you need to do to get apache / mysql running for testing is press start, then its really not needed i guess.
So its just for convenience really in simple terms, there are more technical aspects of services and processes, but thats really not needed at this point