Well I'm no chemist (and thats what it would take to explain things fully) but I'll try my best to explain one thing for you.
In red wines tannins are extracted from the skins and seeds during fermentation. They are considered to be "short chain" tannins when their first released into the wine when its just been made. They can taste very strong and harsh then because they are so small there can be alot of tannins get right down deep around your taste buds. You can't help but taste them really easy. As a wine ages the tannins can start to link together and form "long chain tannins". You can still taste them but since they can't work their way down deep around your taste buds they aren't as overwhelming as they where at first.
I'm sure theres alot more going on with all kinds of chemical bonds joining and releasing as a wine ages. I'm just happy to know how to make wine and keep sampling it so I can see a wine from first made all the way thru its peak and into old age(well some wines make it to old age).
Im n o chemist either and will just add a few more things that will happen. If you have oaked your wine the oak will blend into the wine better and not be so dominant. If you have made a fruit wine some of the harshness of the abv will slowly tame down and the sweetness and flavor will start shine through.
Even chemists don't fully understand the aging process of wine. I agree with the comments of the previous posters, but I just thought I'd throw in my two cents as well.
In addition to tannins softening and oak integrating, primary fruit flavours will generally give way to secondary and tertiary flavours (ie. wine will become less fruity and give rise to more complexity and often to earthy, nutty, leathery notes.) This is important to keep in mind for wines that are meant to be drunk while fresh and fruity (eg. rose, Moscato, Gamay, etc.). These styles of wine might benefit from a few months of bottle age in order to integrate and overcome any bottle shock, but should not be aged too long, or they'll lose their chief attribute.
Aging is essential for some wines to reach their peak and it's always interesting to squirrel a couple bottles away for a few years to see what happens with them; but don't be mistaken into believing that age is going to improve all wines... some are best fruity and young. A general rule is that the higher the level of tannin, acidity and overall extract, the more aging potential the wine will have.