"저의 친구들은 저보다 친구가 많아요."
Translation:My friends have more friends than I.
Technically, grammatically speaking, "I" is correct, since the full completed sentence would be "My friends have more friends than I do/I have friends. (It wouldn't make sense if you said "My friends have more friends than me have friends".) The sentence is comparing the fact that your friends have more friends than you have friends. Thus, it wouldn't be appropriate to use an object pronoun ("me") here. However, in colloquial speech, I think it's accepted as ok grammar.
To expand further though, sometimes this I vs me debate can be ambiguous.
Take the example:
Sara likes cake more than me.
Depending on the intended meaning of the speaker, it could mean:
Sara likes cake more than (she likes) me. (comparing "cake" and "me")
Sara likes cake more than I (like cake). (comparing "Sara" and "I")
See the difference?
Actually the "I" in this sentence is the object of the preposition. (it compares "I" to "My friends") and therefore they should have used "me". The sentence you describe (My friends have more friends than I do), although acceptable in everyday speech, is technically a run-on sentence with two subjects.
This thing is something I've researched before, and a similar example is also discussed online here: https://www.grammar-monster.com/lessons/than_I_me_than_he_him.htm
Both are acceptable in speech. I'm not sure where you got the run on from. I'm open to discussion though.
Here's another one I found