It depends on your threat model. A TPM has multiple purposes, but the most common purpose is measured boot. That is, a TPM will verify the integrity of the BIOS, option ROMs, bootloader, and other sensitive boot components so that it is able to detect an evil maid attack or modified firmware. If your threat model includes an adversary which is able to modify firmware or software on your computer, a TPM can provide tamper-evidence to ensure that it will not go undetected.
So how does a TPM work? It's actually pretty simple when you get down to it. The TPM measures the hashes of various firmware components* and stores the hashes in registers called PCRs. If the hashes all match a known value, the TPM will unseal, allowing itself to be used to decrypt arbitrary data. What data it decrypts is up to you. Most commonly, it is part of the disk encryption key. Unless every piece of firmware and boot software has the correct hash, the TPM will not unseal and the encryption key will not be revealed. TPMs can be used for a lot more, but the idea is the same.
* Technically, the TPM is passive and cannot actively read firmware, bootloaders, or other data. Instead, a read-only component of the BIOS called the CRTM sends a hash of the BIOS to the TPM, starting the chain of trust. This component is read-only to ensure that a modified BIOS cannot lie to the TPM about its hash.
Remote attestation is not something you will likely need to use. It is however not at all unsafe. All it does is allow a remote device to prove to the appraiser that the firmware and software it is running matches a known-good hash. It does not allow remotely controlling the machine. It is up to the OS to do the remote connections and send the data to the TPM. The TPM itself isn't even aware that it is being used for remote attestation. In fact, remote doesn't even have to mean over a network. There are very clever implementations that use a TPM to remotely attest the computer's state to a secure USB device! There are no privacy issues with a TPM's unique private key either due to a TPM's ability to sign things anonymously using DAA, or Direct Anonymous Attestation.
The overreaching issue is that a TPM can prove both to you locally, and to a remote server (with the OS handling the networking, of course) that your computer is in the correct state. What counts as "correct" hinges on whoever owns the TPM. If you own the TPM, then "correct" means without bootkits or other tampering. If some company owns the TPM, it means that the system's anti-piracy and DRM features are fully functional. For the TPMs in PCs you can buy today, you are the owner.
