Why do we rectify then invert current in induction heaters?
For induction heating you want to work at a higher frequency than mains 50/60 Hz.  As it says in that first instructable:

Induction heating works basically like a transformer inducing so called Eddy currents into a conductive workpiece.
This works at all frequencies but in the ragne of 20-250kHz ferrous metals heat the best.

So by rectifying and then re-oscillating you get to pick the frequency you run at, so you can pick a frequency that works most efficiently with the target material.  Better frequency match = more heat for your current = cheaper to run, gets hotter faster, etc.