Exposing wine to air does two things: it triggers oxidation and evaporation. Oxidation is what makes an apple turn brown after its skin is broken, and evaporation is the process of liquid turning into vapor. Wine is made up of hundreds of compounds, and with aeration, usually the volatile undesirable compounds will evaporate faster than the desirable, aromatic and flavorful ones.
There are a few particular compounds that are reduced with aeration, such as sulfites, which are added to wine to prevent oxidation and microbial activity but can smell like burnt matchsticks, and sulfides, which are naturally occurring but can remind you of rotten eggs or onionskins. Ethanol is also a highly volatile compound, and a wine that smells too much like rubbing alcohol when you first open it might lose the ethanol note and become more expressive with some aeration.