Basically, the current in a circuit is dependent of the data manipulated: changing a value from 0 to 1 or 1 to zero generates a current to (dis)charge the gate capacitances.

Maxwell's equations state that a current generates an electromagnetic field, and this field is perceived by the antenna. The attacker is then seeing electromagnetic waves related to the data manipulated.

By carefully comparing the waves with waves where the key is known, the attacker can then guess the key bit by bit.

This is an excellent summary.

Could this be used to break my existing hard drive encryption, or does it only apply to the key generation stage?

In theory, yes. In practice, just grabbing your unlocked laptop and running off is a lot simpler than our/my colleagues' attack. ;-)

A "mind if I share that table" attack might be much more useful than explicitly taking some laptop and running away with it. It completely depends on your threat model.

it can read when the keys are used, i believe, but the attacker would need to know when the keys were used to identify the right time. However at 30cm this is someone standing at your desk waiting for you to fire up an instance or something.