How I understand it is:

• You go to your bank (or use a webapp or whatever) who knows who you are and get them to initiate a withdrawl from your bank account to your Taler wallet in the amount of, say $100.
• The balance in your Taler wallet goes up by $100. The bank also decrements your bank account by $100 and puts that $100 in an escrow holding intending to pay it to whatever recipient(s) can provide cryptographic proof that you gave them Taler.
• You go to a merchant and pay out of that $100 Taler balance $9 for a cheeseburger and fries.
• The merchant receives $9 in Taler from you and checks with your bank that that $9 hasn't already been spent previously before concluding the payment process and giving you your receipt and burger.
• You now have a burger and fries and your Taler balance is $91.
• But the merchant doesn't learn anything about your identity in the process. But they do have proof that your bank has $9 in escrow earmarked for them (the merchant) specifically.
• And your bank doesn't know which of their customers to which they've ever given Taler is the one buying from the merchant in question. They just know that of the total sum of Taler they've issued that hasn't been collected yet, $9 is earmarked for such-and-such merchant/burger joint.
• The merchant can settle up any time, but theoretically the bank can charge per-transaction fees. In order to minimize fees, it behooves the merchant to batch up settlements. The merchant can claim actual USD for every dollar that was used at that establishment by customers via Taler over, say, the last week or whatever in one big settlement batched transaction.

I'm leaving out some details, but that should give you a decent idea of how things work with Taler.

Now, as for this bit:

if Taler can change hands during said "link".

That, I'm not sure of. It might be that you can transfer Taler from your wallet to someone else's wallet (that they could then spend) without any identities being revealed, though they wouldn't be able to get real USD or whatever without working with your bank which would generally insist on confirming their identity. But I'd think in order for the recipient in that situation to know that they actually had real Taler and not Taler that you had already spent and that wouldn't actually work if they tried to spend it or cash it in, they'd have to make basically an API call to your bank, though unless the bank blocked all traffic from every VPN and traffic anonymizer (like Tor or I2p) in existence, I see no reason why it couldn't be done in a way that preserved the recipient's anonymity.

So yeah. Not sure. But even if that bit isn't a thing, I still want Taler to take off.