The layperson answer is:
You are making air expand in a contained space but giving it an escape path in a particular direction that is easier to take than making the container blow up. When I was a kid and it was possible for kids to buy firecrackers from the corner store, I used to like to make rockets out of them to propel balsa gliders. You rolled the end of the firecracker between your fingers to make the packing fall out so it was open at one end, and taped it to your glider. When the fuse burned down and the gun powder lit off (just after you threw the glider), the expanding gases went out the open end, making a nice rocket that ran about a second. Close off the open end, like a normal firecracker, so the gases have nowhere to go, and it blew up (which would occasionally destroy my gliders when I didn't roll out enough of the packing - good thing they only cost a dime in 1967).
So air is being made to expand and you give it somewhere to go. Since the front end is open as well, why it doesn't go back out the front is simply because the engine is packing in air from that direction faster and harder than the burning air can push back, relative to its fairly easy path out the tail pipe, so it takes the "easy way out" (jet engines can back flow if things go off kilter - you will sometimes see flames coming out the front during "stall/surge" events).
In taking the easy way out, it's still has to accelerate to high velocity to make it through an opening that is constricted, and there's your Newtonian thrust reaction.
The last part is, you put a windmill, a turbine, in the high velocity flow and use the windmill to drive a compressor to keep the air being packed in at the front. You are extracting some of the energy in the flow going out the turbine section to power air-packing equipment at the other end, to keep the cycle going by itself without any external help. Basically you are redirecting a part of the energy of the burning fuel air mixture to keep the cycle going instead of making thrust.
In a piston engine, you have a similar energy transfer effect, in a radically different format, to self-sustain the cycle. Air burns in a cylinder's combustion chamber and forces the piston down to drive a propeller, but some of that rotation energy is being used to drive the adjacent piston on its compression stroke, redirecting a part of the energy of the burning fuel air mixture to keep the cycle going instead of making torque at the prop.
The jet engine does the same thing in a linear fashion using aerodynamic forces instead of closed spaces inside cylinders, depending on the inertia and velocity of the flow (like a hallway packed with people running in one direction - forget about trying to run the other way) to channel all of the forces where they need to go to keep the cycle going.
