This honesty might be difficult to answer, cause I kinda just had them met and after a short chat, they've been hanging out together.
Friar's more the broody reluctant friend, and Bucky's just friendly to everyone, so "opposites attract" cliché box is ticked.
I suppose, Friar feels sorry for Bucky, given that he's the newcomer and at the lowest social rank. And Bucky likes to listen to Friar's speeches and is curious on his backstory.
So, why does Friar like Bucky? Cause he's not the fighting/dominant type of rooster like the others and sees him as loyal, and optimistic which compliments his broodiness.
Why does Bucky like the preachy, ignored, and barely-tolerable Friar? Guess he's curious about the Order and looks to Fry as a source of information...and he really likes how Friar's eyes light up whenever he preaches.
Maybe as the plot develops, I can dissect their friendship a little further. Still kinda the start of it all...