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The waveforms are predicted by our knowledge of general relativity; specifically what the gravitational wave profile would be for two objects of certain masses spiraling into each other.

A semi-hand-wavy way we know that the objects in this particular inspired are black holes is that for them to be orbiting each other this fast, they must be very close. For them to be this close, they must be very small. Together, this implied the objects have a certain size which is smaller than their Schwarzschild radius, which means they are probably black holes.

It's mainly derived from the frequency and magnitude of the wave. The frequency derives from the period of the binary system, which in turn derives from the masses of the two black holes. Once the period has been determined it's possible to solve for the original masses. Given those we can estimate the energy produced in the collision, compare that to the magnitude of the wave we received, and then estimate the distance (similar to estimating how far away a sound is by how loud it was). LIGO can also roughly locate the direction the waves come from by comparing when the signals are received at the two different detectors (LA and WA). If we add a third observatory at some point we should be able to triangulate much more accurately.

We're pretty sure they come from black holes because the signals are uncannily similar to what theory says signals from black holes looks like. So either they're black holds or the theory is wrong or there's something else that acts very much like a black hole.