How Does Wine Enhance Flavor?

The other answers make good points, but OP in comments keeps asking whether alcohol helps ingredients "release their flavor" more. And yes, it does.

As to how it does so, one reason is simply because alcohol is a good solvent. Many things dissolve more easily in alcohol than in plain water. (Note that alcohols are often used in other household applications requiring solvents, stain removal, removal of other "gunk" when cleaning, etc. Household cleaning fluid can make use of various alcohols -- not just ethanol, as found in wine -- but the chemistry of how most alcohols work in creating better solubility is similar.)

Another comparison to think of is the use of alcohol in creating things like extracts. You'll get more flavor out of a vanilla bean by soaking it in high-proof alcohol compared to plain water. That's the same rationale behind the concept of a "vodka sauce" too.

Obviously wine doesn't have as high of an alcohol content, but the alcohol that is present can help "release flavors" through better solubility, part of the reason why wine is often used for deglazing pans too during cooking. (As mentioned in other answers, the specific flavor components found in wine are also tasty in and of themselves.)

As you are asking how wine enhances the flavor of foods, the first thing that came to mind for me is that wine contains glutamates, which are flavor enahncers.

Most people would be surprised to know how many foods contain naturally occurring glutamates. A table on this page lists many of the foods containing glutamates along with the amounts (mg per 100g).

This article found on Wine Spectator, explains that fermentation increases the glutamate or umami levels of foods.

While many foods have natural amounts of umami, their umami levels can increase when they undergo various transformations. The most elemental of these is the ripening of fruits and vegetables. For example, a ripe tomato has 10 times the glutamate of an unripe tomato. Drying, curing, aging and fermentation all increase the umami level. Dried shiitake mushrooms and dried sardines have considerably more umami than their fresh counterparts. Why does aged beef have more flavor than unaged beef? It has more umami. Fermentation gives soy sauce, Asian fish sauces and many other condiments such as hot sauces, Worcestershire sauce, Vegemite and Bovril lots of umami.

Fermentation also applies to beverages such as beer and wine. Hanni says big, rich red wines, especially those with high ripeness levels such as Australian Shirazes, and whites that have extended lees contact such as "big, fat, ripe, creamy Chardonnays and round, delicious Champagnes" tend to have the most umami.

What many of these methods have in common is that they break down foods into smaller units of flavor, which are easier to detect. These smaller units, says Shirley Corriher, a food scientist and the author of CookWise, The Hows and Whys of Successful Cooking (William Morrow) "make taste receptors go 'ding ding' in our brain and say 'this is good.'"

So, when you cook with wine, you are adding natural flavor enhancers to your dish.

Edit in response to comment:

From a Science Direct article:

Many food ingredients, including monosodium glutamate (MSG), NaCl, and sweeteners have been termed ‘taste enhancers’ but their main effect is simply to add more molecules that generate additional taste or smell sensations. Tastants such as MSG, salt, and sweeteners don't actually boost other chemosensory properties but rather contribute additional meaty/savory, salty, or sweet properties respectively.

It's simply an ingredient, like any other ingredient you might add. It's flavor chemistry, and is perceived by us as taste and aroma. Alcohol doesn't entirely evaporate. It does help with the release of flavor and aroma molecules in other ingredients. Depending on the wine, and how it is treated in your cooking process, it potentially adds the flavor and aroma of the fermented grape, and it adds acidity to a dish. Also, particularly when using red wines, tannins add to the earthy and "dry" flavor and aroma perceptions.