Why is there a new Group for me as a User?
Solution 1:
Being an administrator just opens up a bunch of other options to you as a user (they are mostly unimportant), but it won't really change anything.
By default, the first user (the one you create from the installer) is an administrator (essentially) with some options disabled for safety. The reason everyone gets a separate group is simply because it makes it easier to manage permissions (on a Unix) system in this way.
Solution 2:
This is necessary because of Linux file system rights. Linux (and other Unix variants) have a right setting for 3 types of users: a owner, a owning group and "others" (i.e. someone who's not the owner and not in the owning group). All files (on linux filesystems) are obliged to have a owner and owning group.
Since usually files that are created by a user are meant for that user only the owner is set to the user and the owning group is set to a special group the user belongs to: the group with the user its name. Thats why such a group is created whenever a new user is created.