It's just a convention used for separating logic out of the main .module file.
The [modulename].pages.inc file typically contains page callback functions (those defined in
hook_menu()). Since these functions aren't used in every page request there's no point including them, so they're auto-included from the
file property defined in
hook_menu() when needed.
You'll probably come across [modulename].admin.inc too, which typically contains callback functions for admin pages.
[modulename].api.php is another convention that Drupal core uses (and subsequently a lot of contrib modules use) to show examples of the hooks they provide, and usually how to use them.
There are other [modulename].[group].inc files that can be auto-included when another module implements
Normally hooks do not need to be explicitly defined. However, by declaring a hook explicitly, a module may define a "group" for it. Modules that implement a hook may then place their implementation in either $module.module or in $module.$group.inc. If the hook is located in $module.$group.inc, then that file will be automatically loaded when needed. In general, hooks that are rarely invoked and/or are very large should be placed in a separate include file, while hooks that are very short or very frequently called should be left in the main module file so that they are always available.