Why would eating the meat of cows fed with antibiotics trigger an immune response to the antibiotics?
It's difficult to know how to answer this question because it is unclear why you think that this might not happen. 

As skymningen has said in comments, any foreign molecule is capable of acting as an antigen,  in principle. We know that people develop allergic reactions to penicillin. And if you do a Google search for immunoassay [insert name of antibiotic] you will probably find results indicating that there are commercially-available kits for assaying the antibiotic based upon antibodies directed against it.

It is however worth noting that these anti-antibiotic antibodies were certainly not raised directly by feeding, or even injecting, the antibiotic. This is because the immune system doesn't really respond to small molecules. If you want to make antibodies directed against a small molecule you have to tag it on to a protein - this is called haptenisation. Then some of the antibodies that develop against the modified protein will recognise the hapten (the small molecule). Another example of this is the development of immunoassay for steroid hormones.

So we have to imagine that a fed antibiotic enters the bloodstream and then reacts chemically with a blood protein - essentially the antibiotic has haptenised itself. I seem to recall reading that β-lactam antibiotics are fairly reactive against lysine residues. The propensity of different antibiotics to provoke an immune response may be simply a function of their reactivity.