What exactly is "breaking in" on a new cartridge ?

What they said, plus your ears and brain.

I am breaking in a Ortofon 2M Black

Those (burn-in, break -in) are frightening expressions or phrases considering the fragile nature of the objects involved. In Europe we use the expressions 'play in' or 'warm-up'(time) which are obviously more adeqate for the purpose.

No.

New piston rings, new semi-auto handgun, etc., these
things need mechanical "wearing-in", so they loosen up and
operate with less friction.

What is happening to a brand new cartridge during it's
"break-in" period ? I do not mean "warming up" for
regular daily playing.

Noslepums, if you read my first response, I indicated that all moving parts within the cartridge need to wear in so that they move freely in interacting with each other. I only indicated that a "warm up" period seems to help for each session. But that was not in place of the "break in" period. There are elastimers that hold the coils and stylus
and they are more "stiff or ridgid" when new. So as the record plays those parts "limber up" to move more freely and establish a path of movement that it will assume each time it plays. I think you answered your own question with you last comment. Yes as a piston ring wears to fit the cylinder or the action in a gun wears to slide more easily, a cantilever to which the coils are mounted need to do the same. The suspension elastimers need to limber up to reduce resistance to movement and more accurately track the groove. It has to move not only up and down but side to side as well. An analogy offered is a blind man reading braille with a limber finger rather than a splinted stiff finger.
http://en.wikipedia.org/wiki/File:Moving_Coil_Cartridge.jpg
this cross section may give you an idea of the inner parts
involved that need to "break in".
actually maybe this is a better read.
http://shure.custhelp.com/app/answers/detail/a_id/4072/~/high-fidelity-phonograph-cartridge---technical-seminar