I thought one can directly send bitcoin to another address without the need for a middleman?

Yes, but dollars and euro and yens almost always require a middleman, unless you're doing cash exchange.

Ah, I did not realise that this was about a "transformation" between bitcoin and normal money. But in that case wouldn't it make sense to keep the money on the exchange just as long as needed?

The bitcoin exchanges typically require 6 confirmations of a transaction in the block chain before crediting your account with bitcoins. This takes about 1 hour.

If you are speculating on bitcoins and you hear about some news that might cause the price to drop, then you don't want to wait 1 hour before you can sell them.

Can you offer more money for the verification, to get it done faster? I think the default is 0.01 BTC, so if you offer 0.05 BTC will you get priority?

At the moment even no-fee transactions usually make it onto the block chain as quickly as possible. The ~1 hour (6 block) confirmation time is due to the recipient being cautious that the sender won't try to double-spend the money by rewriting history with lots of computation power. As more time elapses from when the transaction occurred, this becomes harder, and not as a function of your transaction fees.

The transaction fee only helps get it into the first block sooner. It does nothing for the X confirmations after the first.

Of course one can. But what he's talking about are large-scale exchanges, where one makes transactions with funds deposited directly on the exchange.

You can always do that if you trust the person enough. The idea with exchanges is essentially to act as an escrow service, but one where the two parties don't have to directly know who each other is. I don't think you can decentralize trust, realistically. Someone ultimately has to be responsible for a bad transaction or handling of BTC.