How does a coffee roasting machine work?

Coffee receives heat from the roaster in three different ways. Conduction, convection and a little bit of radiation.
There are various coffee roaster designs on the market. The most common being a classic drum roaster. So lets talk about that one.
Here the beans get loaded into the hopper on the top of the machine.
We open the hopper and the beans fall into the drum of the roaster.
The beans get tossed in the hot drum.

Once the beans are done we open the hatch door and they fall into the cooling tray. Once they are cooled sufficiently they are then dropped into a bucket or other vessel, ready for packing.

A classic drum roaster uses a gas flame under a steel drum to create heat in the roaster, the drum spins like a laundry machine, but slower. This insures the beans don’t get baked or get scorched and are roasted more evenly. When the beans are in contact with the drum this heat transfer is called conductive heat transfer.
The slower the drum speed (to a certain point), the more conductive heat the beans receive.

When the beans are off the drum wall and in the air, they are receiving a convective heat transfer. This is where airflow plays a big part as it is the main means of heat transfer. Airflow also makes sure pollutants like smoke, chaff and others are taken out of the roaster during a roast and prevents a smokey effect on the beans flavour profile.

Professional roasting machines are always (or at least should be) controlled with software like Cropster or Artisan.
These softwares are linked to Thermocouples that send temperature data to the program we are using in the form of a graph, this allows us to know in real time what is happening in the roaster and control the way the beans receive heat.

This is the main way these beans get the flavours you are able to taste in your cup.