A proper answer to this question is of necessity a long one, so I will refer you here >>>>> http://www.howstuffworks.com/turbine.htm
which will give you plenty of information.
but an improper answer would go: air is drawn in at the front, compressed, mixed with fuel then ignited. The heat this gives off forces the air to expand, which makes it rush out of the back very fast, pushing the plane along.
Strictly speaking, the thrust is produced by there being pressure (internal) on the front of the engine which is not balanced by pressure on the back, since the gases squirt out through the jet pipe. The engine produces least power in static conditions, e.g. at the start of take-off, and (at least theoretically) most when moving forward at the same speed as the exhaust gases are moving rearward.