Hi,
This does not look like a very good design, at least not at first glance. Here is how it appears to work and why it's not that good.
1. First, one transistor turns on and that turns the other one off if it was on already because the gate drive drops.
2. Next, current starts to flow though the mosfet because the inductor allows it to ramp up. If it were not for the inductor it would shoot up faster.
3. Next, the current reaches the level where the MOSFET can no longer keep it's drain to source voltage low so the voltage across the MOSFET ramps up. Unfortunately, the current stays high.
4. As the mosfet drain source voltage ramps up eventually it reaches the point where the other mosfet turns on, and that kills the gate drive to the first mosfet thus turning it off. This ends one half cycle.
5. The same thing happens with this second mosfet and that's the second half cycle.
The circuit appears to 'work' because the thing oscillates and drives the output. The biggest problem however is that the mosfets have to stay active for the time where the current ramps up (#2 and #3 above) and this will cause an enormous heating effect. Even though this 'linear' period does not last that long in time, in duty cycle it could be a large part of the entire half cycle, thus eating up lots of power and also overheating the mosfets. Fixing this problem will not only mean the mosfets run cool, it will also reduce the input power requirement and allow a smaller heat sink for the rectifiers.
This same kind of circuit could be used with bipolars probably with better success because bipolars will pull out of saturation at a faster rate. In fact, i would not be surprised to find that this design came from an original design almost the same except using bipolars.
There is also a very likely second potential problem: the inductor. The inductor for this kind fo circuit would have to have a large DC current rating so that it does not saturate with DC current too easily. It has to be able to handle a lot of current and still act like an inductor. If not, it acts like a resistor which presumably isnt as good.
We could look at solutions if you are interested, except of course if you want this circuit to double as a french fry or chicken parts deep fryer