It doesn't.
This phenomena could possibly be specific to you if you always have a huge face?

This could be due to the inclusion of 'Vaso-constrictors' with the principal anaesthetic agent. These drugs restrict the local blood-vessels, this is for two reasons;

Prevent premature dispersion of the anaesthetic agent.

Minimise blood-loss.

The swelling (Oedema) of the area around the administration site, is a side-effect resulting from the restriction of these blood-vessels. As the blood-flow is not totally cut off, the anaesthetic and vaso-constrictor are eventually dispersed. This results in blood-flow returning to normal and consequent reduction of the Oedema.

A cartridge of anaesthetic is approx 2 mls so it is an appreciable amount and depending on how much is given, it can produce a swelling. But mainly your mouth feels big because you dont normally notice it at all and when it is numb, the funny feelings as you touch it with non numb parts makes you more aware of it - Your mind equates this with "big"