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

Suspension, coils all moving parts developing a working motion together. I find that my Kleos now well "Broke-in" still sounds better once I play one LP side to "warm up" for each session. YMMV

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

Nosleplums, as you indicate and as does Tketcham, yes you are right. The break in period is the initial wear in period. But once the related parts wear in the rate of wear is dramatically reduced. No different than the rings against a cylinder other than there is no lubricant. As far as the stylus it doesn't really break in to fit the groove. It is harder than the vinyl and played repeatedly will wear the vinyl before any visible wear occurs to the stylus. Eventually, yes the stylus does wear and then you get to buy a new cartridge and start the process all over again. But hopefully that is many many hours of enjoyment later. Just think of it as new jeans or new shoes that need to flex before they are comfortable. The moving parts a just "limbering up" to work more freeley and as mentioned earlier, to more accurately follow each and every nuance of the record groove, extracting as much information as it can.