I have an airing cupboard, a hot water tank and my boiler has a pump.
How/why does the combi boiler get rid of both the latter?

The difference between the (instantaneous) combi and the non-combi boiler is
that the combi boiler has a heat exchanger that heats the incoming cold
water as you need it. It does this without any tanks or hot or cold water
storage. Advantages are that it uses the full mains pressure, so you get
excellent showers (assuming a reasonable water supply). Unfortunately, it
will heat only so much water at a time, so bath filling tends to be slow.
They are a good choice for houses where the bath is used infrequently, as
they are cheaper, smaller and more efficient than a storage based solution.

It gets rid of the external pump, because they almost invariably have an
internal one, making the external one redundant. Even most non-combi boilers
have a built in pump these days.

Christian.