From what I understand, they asked permission from the credited songwriters (of ‘The Lion Sleeps Tonight’) who said they’d be fine with the title but asked the band to in turn cover the original song, which is where R.E.M.'s version came from.
I say ‘credited’ because I think the most well known version of the song is a combination of different contributors over time. It’s based on a song from Africa and then has a set of English lyrics based on the original melody so from one perspective there are different people that could be considered the songwriters. This isn’t unlike ‘Because the Night’ by Patti Smith and Bruce Springsteen but I think both are credited as the songwriters? At any rate, it’s an example of someone taking a song that existed (in some form) but was adapted and became the version most people are familiar with.
Goodness I have a lot to say about a song that I don’t really care for, lol.