How is this procedure a proof of reserve?
What i don't understand is: What purpose has the signing of a selected message by the third party, described at the end of the first page?


Without this step, it is possible that the transfer you're observing is entirely unrelated to Bitstamp. They could have watched the chain for a while, waiting for a transfer of sufficient amount, and then claim that it was theirs.

Assuming you trust the third party is in fact a third party, and is not colluding with Bitstamp, the signature proves that BitStamp in fact has control over the private key the funds were sent to, meaning they also have control over the funds.

Note that there is research around more advanced cryptographic techniques that do not rely on a third party, and don't require moving all funds around. For example, see this paper: http://www.jbonneau.com/doc/DBBCB15-CCS-provisions.pdf.