It depends. Some classes need more fluff than others. Other times, the heavy handed lore is just a complete turnoff. (5th ed Sorcerors, Warlocks, and paladins. for example.) As is some of the fluffy stuff presented in the class introduction. (like you MUST come up with some major event to justify your choice to become a wizard...) That's the sort of stuff I find irritating.
I'm just getting into 5th ed, having skipped 4th ed altogether, and mostly skipped 3.5. Thus, all I have is the 3 core rulebooks, and the 5th ed Keep on the Borderlands book. So far, I've created exactly 3 characters- just to see how things work: a fighter, a Cleric, and a wizard. Those were essentially my favorites from all the past editions, too.
The fighter obviously requires little fluff to explain it, or to tie it into ANY campaign setting- it's self explanitory, and fits everywhere. The extra goodies 5th ed tacked onto it DO make them a tad less bland- but they're still mainly crunch.
Clerics have always been a bit fluffier, as the manifestation presented by the game is a purely fantasy construct. The concept is still pretty straitforward, though: A person chosen by a deity, or someone devoted enough to that deity to be akin to a mortal representative. Someone who is imbued with the ability to use or call on a bit of divine power. TBH, the vast bulk of the fluff should be left up to the player; DM; and campaign. The 5th ed crunch is just a further evolution away from "cookie cutter" clerics, ala B/X or AD&D. I'm still getting a handle on the new iteration of domains- I really like what they did with them, but also liked the 3rd ed version. (esp the larger number of options) The channel divinity ability is also way cool, the way it keeps adding more options as you go up in level. Both of those game mechanics are , imo 50/50 crunch to fluff. And more or less need to be that way.
Wizards are likewise pure fantasy, and thus some fluff is intrinsic. But, again, beyond the concept of what a wizard is- a student of magic, who learns to harness it's might to do all manner of fantastic things- the fluffy bits are best left to the players, DM's and individual games.
The Sorcerer class, is, imo TOO specific in the fluff department. I liked the OG 3rd ed concept better: wild talents who manifested the ability to cast spells organically- without training or study. From there, it was up to the player to flesh it out- as much or as little as they saw fit.
I can, however, see WHY they felt the need to do it. The new magic system almost completely blurs the line between sorcerer and wizard.