I wrote about the potential impact of the most recent Intel ME vulnerabilities a couple of weeks ago. The details of the vulnerability were released last week, and it's not absolutely the worst case scenario but it's still pretty bad. The short version is that one of the (signed) pieces of early bringup code for the ME reads an unsigned file from flash and parses it. Providing a malformed file could result in a buffer overflow, and a moderately complicated exploit chain could be built that allowed the ME's exploit mitigation features to be bypassed, resulting in arbitrary code execution on the ME.
Getting this file into flash in the first place is the difficult bit. The ME region shouldn't be writable at OS runtime, so the most practical way for an attacker to achieve this is to physically disassemble the machine and directly reprogram it. The AMT management interface may provide a vector for a remote attacker to achieve this - for this to be possible, AMT must be enabled and provisioned and the attacker must have valid credentials. Most systems don't have provisioned AMT, so most users don't have to worry about this.
Overall, for most end users there's little to worry about here. But the story changes for corporate users or high value targets who rely on TPM-backed disk encryption. The way the TPM protects access to the disk encryption key is to insist that a series of "measurements" are correct before giving the OS access to the disk encryption key. The first of these measurements is obtained through the ME hashing the first chunk of the system firmware and passing that to the TPM, with the firmware then hashing each component in turn and storing those in the TPM as well. If someone compromises a later point of the chain then the previous step will generate a different measurement, preventing the TPM from releasing the secret.
However, if the first step in the chain can be compromised, all these guarantees vanish. And since the first step in the chain relies on the ME to be running uncompromised code, this vulnerability allows that to be circumvented. The attacker's malicious code can be used to pass the "good" hash to the TPM even if the rest of the firmware has been tampered with. This allows a sufficiently skilled attacker to extract the disk encryption key and read the contents of the disk.
In addition, TPMs can be used to perform something called "remote attestation". This allows the TPM to provide a signed copy of the recorded measurements to a remote service, allowing that service to make a policy decision around whether or not to grant access to a resource. Enterprises using remote attestation to verify that systems are appropriately patched (eg) before they allow them access to sensitive material can no longer depend on those results being accurate.
Things are even worse for people relying on Intel's Platform Trust Technology (PTT), which is an implementation of a TPM that runs on the ME itself. Since this vulnerability allows full access to the ME, an attacker can obtain all the private key material held in the PTT implementation and, effectively, adopt the machine's cryptographic identity. This allows them to impersonate the system with arbitrary measurements whenever they want to. This basically renders PTT worthless from an enterprise perspective - unless you've maintained physical control of a machine for its entire lifetime, you have no way of knowing whether it's had its private keys extracted and so you have no way of knowing whether the attestation attempt is coming from the machine or from an attacker pretending to be that machine.
Bootguard, the component of the ME that's responsible for measuring the firmware into the TPM, is also responsible for verifying that the firmware has an appropriate cryptographic signature. Since that can be bypassed, an attacker can reflash modified firmware that can do pretty much anything. Yes, that probably means you can use this vulnerability to install Coreboot on a system locked down using Bootguard.
(An aside: The Titan security chips used in Google Cloud Platform sit between the chipset and the flash and verify the flash before permitting anything to start reading from it. If an attacker tampers with the ME firmware, Titan should detect that and prevent the system from booting. However, I'm not involved in the Titan project and don't know exactly how this works, so don't take my word for this)
Intel have published an update that fixes the vulnerability, but it's pretty pointless - there's apparently no rollback protection in the affected 11.x MEs, so while the attacker is modifying your flash to insert the payload they can just downgrade your ME firmware to a vulnerable version. Version 12 will reportedly include optional rollback protection, which is little comfort to anyone who has current hardware. Basically, anyone whose threat model depends on the low-level security of their Intel system is probably going to have to buy new hardware.
This is a big deal for enterprises and any individuals who may be targeted by skilled attackers who have physical access to their hardware, and entirely irrelevant for almost anybody else. If you don't know that you should be worried, you shouldn't be.
 Although admins should bear in mind that any system that hasn't been patched against CVE-2017-5689 considers an empty authentication cookie to be a valid credential
 TPMs are not intended to be strongly tamper resistant, so an attacker could also just remove the TPM, decap it and (with some effort) extract the key that way. This is somewhat more time consuming than just reflashing the firmware, so the ME vulnerability still amounts to a change in attack practicality.