Well this is still breaking news to many of us though the issues of Intel's Management Engine have been raised for years & there have been previous problems with it. So this answer may need updating as more information becomes available.
What are the attack vectors ? (physical access to the computer required ?)
In this particular case, yes - physical access is required. However, you should note that one of the purposes of the ME is to allow remote access. It contains a full network stack and even a web server.
What computer (and configuration) are affected ?
Anything with an Intel processor going back quite a few years. Originally, the ME was built into the North Bridge and later integrated to the processor (ref)
Broadwell, Skylake, and newer are affected. Systems older than that do not support DCI (the protocol for accessing JTAG over USB). While these systems do have an ME, it is not accessible over USB
How to protect and/or mitigate such attacks ?
Burn the tracks for the USB off the motherboard! But bear in mind, that only stops this issue not others that may later be found. Don't forget that there are other, actually probably much easier, attacks possible via USB. For high-security use, USB needs to be turned off or give additional protection.
You may be fortunate enough to have a motherboard with a BIOS that lets you turn off AMT. Also, the Wikipedia article referenced above mentions some mitigations for Windows.
Even if AMT is turned off, e.g. in a BIOS setting, DCI over USB can still hijack the ME on vulnerable systems.
Using AMD chips and motherboards may also help though I have no actual evidence for that. Certainly Intel have always been very close-lipped about AMT so I am guessing that AMD chipsets don't have it - it is possible they have their own solution though.
An anonymous user suggested in an edit that AMD systems have something similar, called the AMD PSP, but they did not think it can be accessed over USB.