Logging in is not required to use WowCollector, but you will need an account to access all of its features. Without an account, you can view all achievements and mounts in World of Warcraft!
If you are logged in, you will be able to see which of these achievements and mounts your characters have collected.
Logging in also enables the Tracker feature, which allows you to keep achievements and mounts bookmarked for ease of access. Think of it as a to-do list!
More details can be viewed on the wiki.