"What is it computing up there."

Hyperspectral satellite imagery - think ASTER/LANDSAT/MODIS but more modern, for surficial minerals study.

"How did you get it up?"

How else? Paid a rocket company to launch it into orbit after proving various flightworthiness tests and getting various certifications and permissions from relevant gov't authorities.

"How does one usually talk with their satellite. I guess you don't merely have a dish since it's probably not geostationary."

K-band. Don't need tons of power, just a good LoS from ground on your target. And yea, not Geostat, I'm in LEO.