Our immune system does react to horse antibodies, but as with any adaptive immune response it takes some time for the response to develop. In the weeks before our immune response fully responds to the horse antibodies, the infused antibodies can have their effect.
If you then have a second infusion of horse antibodies, the immune response would not only prevent them from having their beneficial effect, but it might case serious adverse effects. This serum sickness (Wikipedia link) has been known for over a century; here is a paper from 1918 discussing it as a well-known issue:
- The injection of horse serum either in small or in large amounts in human beings is always followed sooner or later by the development of hypersensitiveness of the skin to subsequent injections of horse serum. For the development of this reaction serum disease is not essential.
--THE RELATION OF CIRCULATING ANTIBODIES TO SERUM DISEASE. J. Exp. Med. Published March 1, 1918
Although there aren't many modern situations in which horse (or other non-human species) serum is injected into humans, when it is (as with anti-venom, which is generally produced in non-human animals) serum sickness is still a concern.
Serum sickness is a delayed immune reaction resulting from the injection of foreign protein or serum. Antivenom is known to cause serum sickness but the incidence and characteristics are poorly defined. ... The primary outcome was the proportion with serum sickness, pre-defined as three or more of: fever, erythematous rash/urticaria, myalgia/arthralgia, headache, malaise, nausea/vomiting 5–20 days post-antivenom.
--Incidence of serum sickness after the administration of Australian snake antivenom (ASP-22). Clinical Toxicology. Published 2016