If Bitcoin transactions require ~10 min to clear, how can payments get recognized within seconds? Bitcoin Monitor offers to issue a notification moments after a payment is sent to a Bitcoin address. Yet if Bitcoin transactions require ~10 min to get written into the blockchain and confirmed as official, how can payments get recognized within moments?

On the other hand, Bitcoin.network requires payment via Bitcoin before making the blockchain available for downloading. It publishes a download URL seconds after a payment is made. How can the site be sure the payment is legitimate. How does it know the payment has been made only a few seconds after payment is sent?
When you create a transaction, your Bitcoin wallet broadcasts it to a few full nodes on the Bitcoin peer-to-peer network.  Those full nodes quickly relay it to other full nodes, and it eventually ends up at the full nodes run by the sites you linked to.  That's how those sites know the transaction exists.

The full nodes also relay the transaction to Bitcoin miners.  Each miner may choose to include your transaction in a block of transactions.  (Miners usually choose to include the transactions that pay the highest transaction fees per kilobyte of transaction data.)

Miners have to solve a sort-of random puzzle---whichever miner solves it first can include proof they solved the puzzle in their block of transactions and send that block to full nodes.  Those full nodes relay the block to other full nodes and eventually the block ends up at the full nodes run by the sites you linked to (and to the wallet of the person you paid).

The puzzles are very hard to solve, so a block with proof of a solved puzzle is hard to fake.  This means any transactions in that block are considered hard to undo.  Also, the next block has to point to the previous block, so it would then require solving two very hard puzzles to undo the transactions in the block containing your transactions.  And it keeps getting harder and harder each time a new block is created.

As you may have guessed, it takes an average of about 10 minutes of puzzle-solving by all miners before any particular miner solves a puzzle, so it takes about 10 minutes on average for a transaction to be protected.  Up until that point, you could create a duplicate transaction that steals back the money you spent---so nobody trusts high-value transactions until they appear in at least one block (and more blocks are better).

P.S. Newbie questions are always welcome!