There are a few airlines that have scheduled flights to all the popular safari areas. My first flight was with Yellowwings, the rest with Safari Link. Your tour operator arranges it and it's part of your overall cost. Your camp will drive you to and from the airports, and they know your itinerary.
Typical plane is a 12-passenger Cessna (4 rows of 3 seats). On my first flight from Nairobi Wilson to Amboseli, the pilot invited a passenger to sit up front with him (so jealous!)
You are restricted to soft-sided luggage, 33 pounds total including your carry on. They weigh your bags and if you are over the limit, you pay extra (or, sometimes, they will send your bags on a later flight). This is all I brought: