What is the meaning of this phrase?
In this lesson it is talking about 粉丝(fen si) being the transliteration for the word fan. So if you look back at the lesson it says 粉丝 - 给 - 自己 - 起名字， The phrase meaning is, “The fans made up a name for themselves”. 粉丝 fans 给 give 自己 themselves 起名字 a name. 起名字 just means to think up a name/ to give a name to. Hope this helps.
就给自己起了这些名字 - This also is the same thing which just means, “They gave themselves a name” 起名字- to give a name.