How exactly does this technique make it more private? Ok so he's not buying Bitcoin directly from a regular exchange but he's still receiving it to a BTC address of his choice. Wouldn't it be possible to still link the transaction to him same as if he'd bought it from a regular exchange? I've never been interested to learn about mixers or other anonymization techniques so I could be missing something here.
TLDR: The XMR.to Bitcoins received in the last step are free of any taint accrued by the original Bitcoins sent to Shapeshift.The XMR intermediary point makes them anonymous, because a key feature of Monero is that transactions are unlinkable.
That feature is inherited from Monero's Cryptonote protocol, which uses ring signatures and stealth addresses in a type of zero-knowledge (homomorphic) encryption.
Mixers are not zero-knowledge; they only obfuscate, always leak information, and may be unraveled given sufficient scrutiny. They require trust; Monero does not.
You're in for a treat; start here:
https://downloads.getmonero.org/whitepaper_annotated.pdfPS
PT's technique can be perfected by transferring (with a healthy mixin) the xmr between two separate wallets you control. That ensures your privacy even if both Shapeshift and xmr.to are compromised (or even full blown, cooperating adversaries).