In all my research, I have yet to find anyone who can say what causes allergies. Often, IgE allergies are genetic- if you have allergies, your children are more likely to have them too (food, environmental, etc.)
For intolerances, it can also be genetic, or it can be caused from damage in the gut (digestive track). See my link on Leaky Gut for more information.
For babies- a common cause for food intolerances is the mother's digestive health before, during, and after pregnancy. Use of antibiotics, steroids, and NSAIDs can have a devestating effect on your digestive health. Pregnancy is hard enough on your body, and when you throw in antibiotics, you immune system just gets overloaded. C-section babies are at further risk, because they don't pick up the initial bacteria from mom (normally picked up while traveling through the birth canal) to colonize their gut. And top it off with antibiotics during/after labor, they end up with no healthy bacteria to fight off the bad bacteria. This leads to damage, and ulitmately food allergies. See my "Pre-Birth" post under Our Story for how this happened to us.
Also, here's another mom's take on the effects of c-section, antibiotics, etc. and how they relate to allergies.
Vaccines can also contribute- they are very hard on an immature immune system, not to mention that many are cultured on animal products, eggs, or other high-allergen substances. They also upset the Th1/Th2 balance in your body, so your immune system starts attacking things that are benign.
Can you outgrow allergies? Sometimes, yes. Especially if you have food allergies caused by leaky gut- if you avoid the bad foods, and work on healing your gut, you have a better chance of outgrowing the allergy and being able to eat the food(s) again. Children very often outgrow food allergies. The exception is peanuts/tree nuts and fish/shellfish, which are usually life-long allergies. :(