"Carrier" isn't a mathematical term, they're just using that word in place of the word "set" to distinguish it from ZF sets. "Group" refers to a specific kind of mathematical structure[0], but it's just given as an example. The important part is that any object can be part of a group, so if the collection of all groups existed as an object, then it would also be part of a group, indirectly containing itself and leading to paradox.
[0] https://en.wikipedia.org/wiki/Group_(mathematics)