The drainage pipe underneath is connected to this tiny opening. Many times when using, people forget to turn off the faucet (for instance, when they are waiting for the sink to fill before leaving to answer a phone call). In these situations, the water will stop overflowing onto the floor and filling the bathroom when the water level reaches two-thirds of the sink and drains through the hole into the pipe.
This overflow hole is useful, but it can also be inconvenient for users. Its fairly uncomfortable location makes it prone to bacterial and mold growth, which can cause foul smells to emanate from the pipes.
Thus, in the event that you detect an offensive odor in the restroom, particularly emanating from this tiny opening, you can combine vinegar with baking soda and place the mixture into the hole of the sink.