What I'm about to say is not supported by any scientific evidence that I know of. It comes directly from other discussions I have been a part of as a competition grower. In certain arenas, this can become a very heated topic similar to religion or politics.
Compost tea is a touchy subject. As you can see, there have already been several brewing methods and recipes posted. In the scientific community, I think the jury is still out. I know in the past, commercial growers were advised against its use because of the danger of contamination by anaerobic bacteria (primarily e-coli), especially if foliar feeding, which is the most common application of compost tea. That danger is increased if manure is used in an anaerobic brewing recipe. The most common method of anaerobic brewing is the dump the manure in a bucket of water and wait a couple of weeks method. Daily stirring will not provide enough oxygen to prevent e-coli from forming.
Anyway, the benefits are argued by many.
The proponents argue that properly brewed aerobic compost tea applied as a drench provides beneficial microbes to the soil. They also believe that foliar feeding provides additional nutrients to the plant, if done at the correct time of day. The theory is, during the warmer parts of the day when the stomata are open, the nutrients can enter the plants leaves directly, bypassing the root absorption process. It is also believed that the microbes applied to the leaves act as a defense against disease.
Opponents believe that the beneficial microbes die as soon as the water from the tea evaporates. They also believe that any nutrient taken in by foliar application is minimal at best. Their belief is that the CEC of the humus in the compost will not release the cations simply because it's emmersed in aggitated water. Of course, that only applies to Calcium, Potassium, and Magnesium, it says nothing of other nutrients that may be present such as Boron or Zinc.
I have used compost tea in the past and tested it (most unscientifically mind you). I only ran the test on pumpkin plants, as I will not take even the smallest chance of contaminating my families food. I gave some plants both drenches and foliar applications, while others received no compost tea. The compost tea treated plants did not outperform the non-treated plants in growth or production, but I did discover one thing. The treated plants suffered a great reduction in the severity of powdery mildew. I don't attribute this to the beneficial microbes, but rather an altering of the pH level of the surface of the plant, making it a hostile environment to the fungus.
If you want to make compost tea, here are some pics of a brewer I made years ago. It worked very well.
If you are going to foliar feed vegetables, I would recommend a mix of compost, alfalfa meal, molasses and Alaska fish oil. If you are going to use compost, make sure it doesn't contain manure. The aerobic brewing system should probably prevent the formation of e-coli, but I can't guarantee it.