The 2 biggest things;
Engine clearances - There are lots of areas where clearance comes into play, but what REALLY matters during break in is the cylinder bore to piston clearance (The amount of space in between the piston and the block its traveling up and down in). In older engines the machines that MADE the engine (Boring/Honing machine/Lathes, etc) were not as precise as the machines used today. That means clearances had more tolerance(wider range of outcome in size), and needed to be broken in carefully incase clearances were a bit too tight, and the piston needed to "wear out" the cylinder bores a bit. Nowadays with everything computer controlled, engines are built EXACTLY to the specs they need to be, so really the only issue is the surface of the cylinder wall wearing into the piston ring.....
Cyclinder Wall surface - A brand spankin new engine has just been assembled after machining. This means that the cylinder wall surface is rough on a microscopic level. This is due to the way the machine that bores out the hole in the block does it's job. This surface meets against the piston rings as the piston travels up and down. Basically, by "getting on it" you push the piston rings out into the cylinder wall and help smooth out the roughness created by machining. Smoothing this surface out creates a better seal, meaning less blow-by (When combustion pressure pushes PAST the rings). Less blow by means more power, easier starting, and less emmisions.
So does this mean BEAT THE LIVING CRAP out of your bike when you get it? No...
Metals change over time with heatcycles. Friction creats heat. So the piston rings and cylinder walls wearing into each other get very hot, not only because of the combustion, but because of the friction created between the two.
So how do you wear the cylinder walls/pistons rings without overheating them?
The best way is to get on it hard for 10-15 seconds, then leaving it at high RPM's, decelerate. Lots of oil is spread on the cylinder walls under deceleration, and oil is a very good conductor of heat. This will also help "wash away" small metal particles left behind from the components wearing in. This is also why you should change your oil after 20 miles or so, as most of the break-in is done, and there are metal particles in the oil.
Motoman describes this in better detail, but I hope I put this simply enough that everyone can understand.