A "snapshot" of the Blockchain is taken at a certain blockheight, and a giant file (snapshot.bin) is created that lists all the addresses along with their balances.  This file can be audited by anyone to confirm its accuracy.  This file becomes Block 0 for the spin-off.   

We are debating the fine details, but staking your claim could be as simple as opening up your bitcoin wallet, and producing a bitcoin-signed message of the ASCII string:

    "claimto "

The value of spinoff_address could point to an address you control in the new network, it could be your deposit address at an exchange, or it could be an address controlled by someone you want to sell your spin-off too.  You would then copy the signature produced by your bitcoin wallet and paste it either into your spin-off client, or to a web-based "spin-off TX push service."  This would push your claim onto the spin-off network and assign your share of the spin-off premine to the address you specified in the ASCII string that you signed. 

Does this make sense?