Alright guys, sorry for the noob-looking account, I lost my old account (hence the bad links)

I'm looking to setup a transparent proxy, which (if I understand correctly) will allow me to monitor/control http traffic on my home lan with the use a log analyser.

I'm planning on following this guide:

I'm cheap and don't wanna buy another NIC.

My question: How does this all work? I get that http traffic goes to my server first, and then to the destination address, but how? What is stopping the other computers on my network from going straight to my router?

Is my interpretation of a transparent proxy correct?

My proposed setup:

I know it's a broad question, but I'm just genuinely interested.