Simply exposing wine to air starts two processes. It starts evaporation and oxidation. (1) Evaporation is the process of liquid turning into vapor, and (2) Oxidation is what makes an apple or banana turn brown after its skin is broken. Wine consists of hundreds of compounds, as air is introduced, most of the undesirable compounds will evaporate quicker than the desirable, aromatic ones.
Specifically, the components you want to get rid of are the sulfites which are added to wine to prevent oxidation and the naturally occurring sulfates that are created during fermentation. Both can smell like rotten eggs or propane gas. Another bad smelling compound that is dissipated by aeranization is Ethanol, it can smell like rubbing alcohol which has obvious negative impact on the overall aroma and taste of the wine.
Just by opening a bottle and pouring a glass you will to a degree aerate the wine. Swirling wine in a glass before tasting again adds more air to the wine and helps evaporate and oxidize away some negative qualities of the wine. And of course, decanting works the same way. But for fast and complete results, an aerator speeds up this process by spinning and mixing air evenly through every drop of the wine. The heavier and more concentrated the wine the more it will improve through aeration and the longer the effects of aeranization will last.