It's all really elegant.
There's a set of satellites wandering the heavens, and each knows exactly where it is and what the time is. Each constantly transmits this information at the speed of light on certain radio frequencies.
You have a receiver tuned to those satellites. Assume your receiver also knows what the time is. When it picks up a signal from a satellite saying "I am at position Z and I sent this at time T", your receiver can say "I got this at X seconds after T, and since radio waves travel at the speed of light -- C miles per second -- that means the satellite's X times C miles away from me. But I've no idea in which direction, so I'm somewhere on a sphere X times C miles away from position Z."
That works with one satellite. If you repeat the trick for a second satellite, the receiver knows it's also somewhere on a second sphere centered on the second satellite. Imagine the two spheres colliding -- the receiver has to be somewhere along the circular line where they meet. Add a third satellite and a third sphere, and you're down to a couple of points.
How accurate all this is, is the primary concern of GNSS. If you add a ground-based transmitter that knows exactly where it is, it can monitor the satellite transmissions and spot when the timing's drifting a litte -- it can do this because of atmospheric conditions or small errors on the satellites themselves. This ground based transmitter can then be picked up by your receiver and used to correct the signals you pick up from the satellites.
There are other issues, like how you make sure your receiver has a clock good enough to do all this accurate timing and how you spot various errors that look like they're not errors. And how you get all that in a handheld unit that costs less than a hundred quid and runs for ages off batteries (that's the bit that really impresses me!)
R