When I was in school, allergens were taught as follows (This also explains why people presumed it would fail):

1) Your body "sees" this, meaning it has antibodies that react to it

2) Cells whose antibodies are activated replicate, as they are recognizing something that is, to the body, presumably foreign and bad

3) Because you now have more cells after the originals replicated, the effects of the allergy worsen as you are now more reactive than before

I'd be interested to see some tests on this. I have a few hypothesizes that we could test:

1) If the children are still reactive to immunological tests (Meaning they still have the antibodies!), are those antibodies being blocked by another compound, another cross-reactive antibody? Another idea is that the actual immune response to antibody activation (Recognizing peanuts) might be handled differently within the cells that produce the actual response to the allergen (Swelling and stuff).

2) If the antibodies no longer exist, I would start looking in the areas where immune cells develop to see if cells producing reactive antibodies to the allergen are being killed off during development (This process is also why you don't kill yourself with antibodies recognizing yourself). If that isn't the case we need to track individual immune cells over the course of treatment.