The ICs are for functions like this:
-Switching power supply control (main IC)
-Power transistors for the actual switching power supply work
-Input-side voltage regulation, as needed
-(Possibly) voltage feedback photocoupler
You will typically find a large fuse on the input side of a step-up inverter (which is what you have) because mathematically we must supply a larger amount of current (at low voltage) to supply an equal amount of power to the output side (high voltage, lower current).
The fuse is to protect the switching power supply and supporting circuitry.
You have to remember that unless you have an expensive "true sine wave" inverter, it's not actually a sine wave but just high/low pulses used to average out to what an AC load would get & use.
The problem is that the output is not a real sine wave, so many devices don't like it, can't work right, and it's bad in some cases for the load (as the other posters said).
(That's one reason you see a lot of complaints about people having electronics that won't work right with "AC inverters")
Unfortunately a large number of inverters are of very low quality and questionable reliability, at best!
I've designed a few inverting power supplies, used a few low-cost DC-AC inverters to get 12V-260V for a car vaccum tube audio project. :-)