Where does this wire connect?

Okay, so those are the wires that transmit the sound to the ear speaker. It looks like there are only two wires, which would make sense since you just need a plus and a minus (+ and -) connection to power the speaker.

On your other picture, there's an obvious place where two wires have been pulled off the circuit board; I've circled them in your photo.

There's no indication about which of the two wires goes where, but generally a red wire is the plus (+) side, so the other (brown?/yellow?/tan?) wire will go on the minus (-) pad. If you can find the wires leading to the speaker for the other side, that may tell you which is which for sure, assuming they use the same color coding. Otherwise, try soldering the red to plus and the other to minus and see if it works. If not, you may have to swap them around.