Actions in D2 like attack, casting a spell, blocking, or taking hard hit (hit recovery) have animations. The whole game runs in frames - there are 25 frames in one second. So for example a simple attack animation with a short sword for paladin is 14 frames long. But with some Increased Attack Speed (IAS), it can get faster and it can be only 13 frames long (with more IAS 12 frames etc.)
Breakpoint is minimal amount of IAS needed to get faster animation.