so, please if anyone can help me, what is the variator and it does what?
and is it a serious spare part, and to replace, is it cheap? or it cost alot to change?
The variator's job is to adjust the timing of the engine valves according to how fast the engine is running (the profile of the lobes on the camshaft governs this normally, and the variator, in effect, adjusts this profile - hence "Inlet Cam Variator"). When it fails, the engine timing stays in its "high speed" setting, which is exactly the wrong timing for a cold engine starting up. The sound you hear is the engine "knocking", as a result of getting too lean a fuel mixture (valves fully open = more air in the mixture).
In answer to your second question, yes, unfortunately this is a relatively expensive repair. It's normally done with the timing belt, but requires a bit more delving into the engine than that. I've never had it done on its own, so I couldn't give you an idea of price, but doing the variator usually adds another €300/£200 to a belt service.
It's possible your seller got just the belts done, were told about the variator, but thought that as they were going to sell the car anyway, it wasn't worth the cost.