Why did Murray invite Arthur to the talk show?

I doubt that the reason behind Murray's inviting Arthur was to help him or get famous.
He clearly was mocking him. For example when he implied that Arthur laughed in his stand up comedy performance because he thought laughing so much would force the viewers to laugh too (suggesting that he was bad at jokes so he chose this as an alternate way to entertain people instead)

Also, Murray's face expressions clearly explained that he was taunting Arthur through every word that he spoke, and that he was not being amused by any of his jokes. He had just called him to increase the ratings of his show by interviewing a person who had gone viral because of his weird behavior. (Laughing previously during his performance instead of doing comedy)

The movie does not answer this (and indeed, does not even make clear that Fleck was ever even invited, this may have been entirely in his head).

I prefer to believe that the invitation was real and that Murray’s motives were not merely to make fun of Arthur. Showing the clip of the lousy performance and then having Arthur give another lousy performance live would have been bad television, and Murray is, if nothing else, a good television host.

So I like to imagine that he was hoping that Arthur could defy the studio audience’s expectations, have a successful (or even semi-sucessful) routine, and have people cheering for his bravery at taking the stage. A little bit like the William Hung experience on “American Idol”. Hung was a bad singer, but his bravery at taking the stage and his good spirit about the audience’s reaction soon won them over. Murray was, I think, hoping for something similar.