Letting a wine ‘breathe’ is one of those confusing terms that winos like to throw at people - quite unnecessarily, because the process itself is actually very simple. What people are talking about is opening a bottle of wine in advance of when you drink it and either leaving it in the bottle with the cork out (or screwcap off), or decanting it into a decanter or jug and leaving it for about an hour or so. 

Why do you want to do this?
If you think about what happens if you cut an apple in half and leave the cut pieces exposed to the air for half an hour – it turns brown and slightly sweet. This is the process of ‘oxidation’ where the oxygen in the air reacts with chemicals within the fruit and starts to cause it to rot. Exactly the same thing happens with grapes if they are split open at any time, both before they are crushed and during the crushing itself. To counteract this, most grapes are preserved with SO2  or dry ice and so oxygen is generally excluded, particularly in white winemaking, in order to keep the pure fruit flavours. This is referred to as ‘reductive winemaking.’

But oxygen can be good too.
Although oxygen at an early stage can be harmful to wine, it is incredibly useful when it comes to drinking it. By swirling the wine in the glass or sucking air in through your lips when holding a mouthful of the stuff, you are allowing the oxygen to mix with the flavour compounds and make them easier to identify and taste. Oxygen also accelerates the ageing process so if you are trying a young red wine, it can soften the tannins and make it easier to drink and a white wine can gain added complexity from being opened for a short while before being drunk.

Source:Food24.com