Electrons move freely through the wire until they reach the light bulb's filament.
Imagine that you're holding a garden hose -- one with no nozzle attached. With nothing to obstruct the water, it pours out of the hose's end freely. But if you place your thumb over the end of the hose, the water's going to squirt out. The reason it does is because of the resistance created by your thumb.
It works much the same way for a light bulb. Electrons move relatively freely through the wire, then they come to the bulb's filament, which resists the flow of electrons.
The electrons can get through, but not as easily as they can through the wire. The work done overcoming the resistance causes the filament to heat up and to give off light.