How can one redeem a Casascius Physical Bitcoin? These physical coins are supposed to be worth 1 BTC each - they have a hologram and a private key that somehow gives access to a real (digital) Bitcoin.

What is the process of claiming this real (digital) BTC for the physical one?
The first 8 characters of the public address are on each coin, which allows you to obtain the full address or check the balance at FirstBits or a similar service. Beneath the tamper-evident holographic sticker on each coin is the private address which corresponds to the public address. Once you've obtained both the public and private address, the process for importing this information into an existing wallet has already been documented in this prior question.

Edit: On a side note, it's probably worth noting that this process is almost identical to the process used to redeem BitBills

Edit 2: Apparently these coins use a different private key format than standard. Casascius has published a utility to convert these private keys to standard addresses and more about the "mini private key" format can be found at the wiki. For those lacking a copy of Visual Studio, I've compiled a copy for your convenience. These keys may also be computed using any calculator or code capable of generating an SHA256 hash:

SHA256("S4b3N3oGqDqR5jNuxEvDwf") = 0C28FCA386C7A227600B2FE50B7CAE11EC86D3BF1FBE471BE89827E19D72AA1D

Of course that's still in hex and needs to be converted to Bitcoin's modified Base58 but many wallet import tools will accept hex as well.

Edit 3: Just heard from Casascius that we may be able to redeem these coins, as well as any device bearing a mini private key directly at MtGox in the near future.

Edit 4: Support has officially been added