How did Coinbase monitor BTC transactions? I've been seeing a lot of articles like this one claiming that Coinbase has been suspending users that transmit and receive BTCs from sites that provide objectionable goods and services (e.g. drug sales, gambling, etc.).

I'm not interested in the why of their actions but the how. 

My knowledge of BTC and Coinbase is extremely limited but I was under the impression that each transaction has a unique target address (or at least it's an advisable security practice). So if a person were to send 10 BTC to "objectionable site A" at newly created address AAA, and then later "objectionable site A" sends 12 BTC to me at newly created address BBB, how would Coinbase even know that address AAA were owned by "objectionable site A"? Isn't it anonymous?

I sort of understand the concept of the blockchain and how everyone can see every transaction that's taken place, but from Coinbase's perspective, it seems like all they'd be able to determine is that someone sent some BTC to address AAA. How would they even know what business AAA is associated with? 

I'm confused. Please explain. Thanks!
Your understanding of Bitcoin is correct.  What you may not know is that the "advisable security practice" of using a unique address for each transaction is not followed by many block chain gambling sites, such as SatoshiDice.  Because these sites reuse addresses, transactions sent to those addresses can be flagged or blocked.

Even if this were not the case, block chain gambling may be easy to detect by analyzing transactions.  For example, I believe SatoshiDice always sends back a tiny transaction for each bet it receives, which is an obvious marker.  There may be more subtle indicators too, like the ratio of spends to receives being near the ratio of the amount spent and received.  Also, Coinbase could have access to non-block-chain info like people's Coinbase addressbooks where they write "dice, 1:100 odds" or "weed dealer".