Multiple reasons. Some have been outlined above, but I'll add a few more and elaborate.
1. Weather Nerf - Why bother jumping through all the hoops to get +1 or +2 boosts per round, when by switching in one Poke you get instant boosts to your whole team? Rain-boosted Water attacks, Sun-boosted Fire attacks, and Sand's SpD boosts + residual damage all contributed to an environment where BP teams were just outclassed and overwhelmed. It was too easy to break BP chains by OHKOing the individual users.
2. Defog Buff + death of Spin Blocking - It was trivial to get Stealth Rock in Gen IV and V. Hazard stacking was even viable. However, in Gen VI a lot of people don't bother setting up hazards anymore thanks to it being so easy to remove them, and besides Espeon could even make it a liability by bouncing it back. Without hazards in play, BP teams can freely pass between members without taking damage each time.
3. Scolipede getting Speed Boost - Scolipede is a serious upgrade over previous mons in the slot of BP Lead/Speed passer. Only Speed Boost Blaziken is better, but that's because he can start wrecking on his own and start a BP chain if he runs into something he can't wreck. Scoli also has great synergy with both Sylveon and Espeon, as they cover each other's common weaknesses both type-wise and stat-wise.
4. Sylveon and Mr. Mime getting the Fairy-typing- Sylveon gives the team a crucial immunity to Dragon type attacks, notably Dragon Tail, rendering moot the one form of phazing that Espeon and Mr. Mime's Soundproof couldn't block before. This means BP teams no longer have to rely on Smeargle's Ingrain (or, in some cases, not run Smeargle at all) to keep their chains. Sylveon also makes for an excellent BP receiver, as his Pixillated Hypervoice + Stored Power wrecks almost everything in the meta.
5. Special attacks getting nerfed - this, combined with the weather nerf, means there are far fewer attacks that can OHKO BP users even after a couple of Special Def boosts. This also means getting SpDef boosts aren't as crucial anymore, and BP teams are free to also boost Def.