I don't post very often in the Motoring section (and, when I do, it's usually in response to a legal question about driving). This is because I freely admit that I know very little about cars. So the following opinion shouldn't be regarded as anywhere near to 'expert' :-)
When you hit the accelerator, you're increasing the flow of petrol to the engine to provide a 'richer mix' of air and petrol. The only reason that I can think of why the engine should stop when you do this is that the mix is TOO rich. As there's no reason why there should be too much fuel getting through, the obvious alternative is that there's too little air. So, perhaps a blocked air filter might have something to do with the problem? (It might not be the root cause. I once had a Fiesta that suffered problems due to a blocked air filter. Unfortunately, the air filter was becoming blocked because it was getting flooded with oil from an un-related engine problem).
The only other reason I can think of as to why the petrol/air mix might be wrong is that the petrol is being injected at the wrong point in the ignition cycle. This would suggest a fault with the engine timing or the fuel injection system.
Personally, I'd start by checking the air filter ('cos its easily done and air filters are cheap). If that doesn't help, then it might be time to get the fuel injection system checked out.
As I've said, I'm no expert, but I hope that this might give you a few ideas to work on.
Chris