I think there isn't a single meaning, and sometimes it might be a simple brofist, showing respect to one another. But there's also to consider that Bee, in a way, uses fist bumps to read the other person's mind/heart/soul, to feel the other person; he does it with the Raikage.
But for example
When Naruto exchanges fist bump with Kurama (The Nine-tailed fox beast): There happens that the two chakra meld/connect, as a sort of energy transfer. Perhaps it's also more than that, but that's basically to show that we don't simply have brofists in Naruto.
I think I found the chapter but it's not flat-out stated. When Bee first exchanges fist bumps with Naruto, he says he won't train him to control his beast: first because he "made a sortfool out of explanation might have been mentioned inhis rap", but also because when they exchange the Mangafist bumps, if you want
Bee feels the Dark Naruto inside Naruto himself, through them. So unless Naruto solves this problem, he won't be able to proceed to the next stage, i.e. the training with Bee.
After the first meeting with Bee (when Naruto uses the Erotic Jutsu), Motoi does ask him whether they exchanged fist bumps. I can trythink this is a hint to look for itthat, because when Naruto confirms,
Motoi takes him and Yamato to the waterfalls. They ask whether that's the place but Motoi says that that's just the first step: Naruto needs to see the truth first, which is where the Dark Naruto is revealed.
Naruto will help save Motoi from the Squid attack that happens then and Bee trusts Naruto can defeat the Dark one.
Hope that helps.