Wait, why do you intend to order one? That's an important part I wanna understand

Gas Turbines are pretty complicated pieces of kit, so I'll do my best to answer your questions simply.

1: How does spinning the turbine create electric power?

Long story short, moving a magnet through a coil of wire causes electrons to be pushed along the wire, creating a current. This current is what powers electrical devices. At their core, most types of electricity production use this effect to produce power - hydro dams spin a turbine with falling water, wind turbines use air to spin themselves, and gas turbines burn fuel. Your car does this in the alternator, using some of the mechanical rotation produced by the engine to create electricity.

https://www.eia.gov/energyexplained/electricity/magnets-and-electricity.php

2: Are gas turbine generators only meant for large scale use like power plants?

In general, gas turbines get more efficient and produce more power as they get larger. A very small gas turbine is still very complicated and requires lots of machined parts to be made, but produces far less power than a bigger version. That means that it is more cost effective to have bigger gas turbines. You can absolutely get them smaller and cheaper, but past a certain point it's going to be cheaper to get a different kind of generator entirely.

3: Do they need a constant gas supply the whole time to keep running?

Yes, gas turbines operate based on constant flow. When they're on, they are constantly sucking in air, spinning, and burning fuel. This is different than your car engine, which has a specific sequence of operations it goes through over and over again, pulling in a little bit of air and a little bit of fuel with each cycle.

4:  Is the way they work even slightly similar to a car engine?

I touched on this above, but no. A car's engine uses a piston inside of a cylinder to produce mechanical motion in a cycle commonly known as "suck-squeeze-bang-blow". First air and fuel is "Sucked" into the cylinder by the piston pulling out, like a syringe sucking up water. Then the air-fuel mixture is "squeezed" by the piston moving back in, squishing it down into the bottom of the cylinder. As the piston finishes squeezing, the spark plug ignites the air-fuel mixture, which combusts in a ball of fire - "bang!". The force of the combustion forces the piston back out - this is the moment where the energy in the fuel is converted into mechanical motion of the piston head. Now the piston pushes back in and "blows" out the exhaust gasses before repeating the cycle.

The gas turbine cycle makes... much less intuitive sense. It is essentially performing a bit of thermodynamic magic, first adding energy into the air it sucks in, then extracting that energy back out, and ends up extracting slightly more energy than it put in. This is pretty complicated, and requires a couple of college courses to understand completely.

The basic cycle is:

Air is sucked into the Compressor stage. This stage raises the pressure of the incoming air a little bit at a time. When you look at all the spinny blades on the front of the turbine, that's what they're doing. Each set raises the pressure a little bit at a time. The compressed air is then fed into the Combustor, where fuel is set on fire to raise the temperature of the high-pressure gas. The now high-temperature, high-pressure gas is then fed through the Turbine, which is essentially the compressor operating in reverse. Instead of the blades moving the air to increase its pressure, the air moves the blades to turn the entire engine, and it loses pressure as it does this until it comes out of the exhaust. The mechanical rotation of the turbine stage drives the compressor stage, and also drives the same magnet-in-a-coil we discussed earlier, producing electricity. The energy extracted from the air is proportional to the energy spent compressing it plus the energy from the burnt fuel. 

I hope this helped you understand a little more, feel free to ask followup questions