This is just a quick reply and may not be exactly right.

They are just maths. Add them to the positive frequencies and the real part is the signal.

If you multiply by exp(jwt) where w is a larger frequency, then they will both become positive and you will get upper and lower sidebands about the carrier frequency w, as happens in common communications systems.

In a real system the signal is real and with positive frequency but mathematically can always be split into the sum of complex signals at positive and negative frequencies.