Well, it's a bit hard to explain. Basically, your LP gains are affected by your MMR, as compared to what MMR you should have for the League you're in. LP tends to change faster than MMR in either direction (so for every win, you might get 20 LP but only gain 5 MMR, and vice versa). So if you win a lot of games in a row, you get way more LP than you do MMR, so the system clamps down on you because your League is now ahead.
So, by way of example, say, you started at 1400 MMR and 0LP in a division where your expected MMR is 1350. You win a bunch of games in a row, and gain a bunch of LP, so now you're at 90 LP in your division. However, because MMR changes slowly, you're only at 1430 MMR and you're supposed to have 1500 to get into the next division. The system clamps down on your LP gains from that point onwards until your MMR matches what you need, because they don't want your League to get too far ahead of your MMR. You could still get into the next division if you continue playing with a high win rate, but it's going to be harder (and note that in the old system, you wouldn't have any chance to advance until you got your MMR to 1500).