ALTER DATABASE statement not allowed within multi-statement transaction

No, you're not doing anything wrong. I got the same thing. I solved it by breaking the sample up into multiple scripts and running each section of the script sequentially, in its own query window, instead of as one big script. This worked in my case because I'm always running these samples in an isolated VM (not on a production server!) and transaction handling is unnecessary since I'm the only one here.

Looking at the script again today more closely, there is no transaction handling defined explicitly, but perhaps you pasted the script into a query window that already had an active transaction, or created a new query window that automatically added BEGIN TRANSACTION; / COMMIT TRANSACTION; statements.

I also pointed out a couple of other potential gotchas in this blog post.

I agree with @AaronBertrand you aren't doing anything wrong. This wouldn't be the first time I have seen a Microsoft script with a bug in it. Realistically with as many scripts as they publish I would be surprised not to see any.

Specifically the problem is that ALTER DATABASE is not allowed in a transaction at all. You can see the BOL reference here: Transact-SQL Statements Allowed in Transactions

In fact even as simple a script as this fails with the same error.

BEGIN TRANSACTION
ALTER DATABASE AdventureWorks2012 SET READ_WRITE
COMMIT

As Aaron said, remove the transaction handling (or at least the ALTER DATABASE statement from the transaction) and you should be fine.