How is tolerance to an allergen developed?
The allergy shot in itself is a form of therapy called allergen immunotherapy (AIT). In it's essence, we're trying to get someone to develop immunologic tolerance to a given allergen such as pollen, okay! So how does it happen at the cellular level? (Assuming the treatment protocol is known)

Understanding The Mechanism Of Allergen Immunotherapy, as an article is kind of a broad overview. Thus there are phases to AIT that include early, intermediate and late phases. The process begins with a downregulation of mast cell and basophil activity, followed by induction of allergen specific Treg cells, a decrease in Th2 cells, and then a decrease in IgE production with an increase in IgG4/IgA.

Mechanistically, it doesn't look like it's well known how basophils and mast cells get regulated by AIT, but the theory is that the mechanism may resemble H2R (a histamine receptor) receptor activity during venom immunotherapy: H2R becomes highly upregulated after therapy start, and inhibits Fcε-R1 activation of mast cells and basophils, even in the presence of IgE (2, paywall on this one).

They provide an impressive table in Akdis &amp; Akdis 2014 as to the role of immune cells in AIT:



So in the process of encountering the allergen, the immune system produces Treg cells as a result of dendritic cell antigen presentation. An inducible type of Treg called Tr1 is capable of abrogating the response of Th2 and Th1 cells through the secretion of TGF-ß and IL-10. They have a good table for this, too:



There's an effect in there due to TGF-ß exerting its effect on B cells that ends up inhibiting class switching to most other immunoglobulins except mucosal IgA. In conjunction with IL-4, IL-10 is capable of promoting the class switch of preceding B cells to IgG4. So the classes of Ig will largely become IgG4 and IgA in response to the allergen of interest. The B cells will also begin to produce their own IL-10, and in a way become a regulatory type of B cell, or Breg, that propagates the tolerogenic response.

And so the big conclusion: (1) the mast cells/basophils aren't as prone to causing anaphylaxis, (2) the effector cells specific for the allergen have actually shifted to populations of tolerogenic DCs, Tregs and Bregs, and (3) antibody production has shifted to IgG4/IgA from IgE. The net result is due to the repeated exposure to the allergen at some optimal concentration, but the key is they performed the treatment slow enough that the immune system had time to build up tolerance without going crazy. It's also likely that we don't know how long the system stays tolerant for after successful treatment.

It's much more complicated and if you can get past the paywall on source 2 there's really good, detailed info. I didn't want to go into a novel here, but let me know if you need clarification anywhere.