You add heat at constant pressure to the air flowing through the engine. Which means the volume has to increase. Only way balance things out is the velocity of the air has to increase as well.

Ah, there's the key.

If you look at the TS diagram state 1 is the intake. 2 is after the compressor. 3 is after the burners. 4 is the exit from the turbine. Temperature is decent proxy for total energy. You can see the vertical lines of constant entropy. That means the energy is being extracted as work. Note real compressors and turbines the lines are angled slightly which reflects inefficiency.

You'll note the difference between 3->4 is larger than 1->2. The turbine expanding the hot gas produces more power than the compressor does compressing the cold air.

Another related question: what happens if you don't rotate the engine before you ignite it?

But jet engines can and do backfire spectacularly at times. You can find lots of cool footage from hobby turbine engine enthusiasts on the tubes.