Case Study: This is atopic dermatitis and can be caused by PCN as well as sulfonamides. I would tag this child's ID band & chart Drug Allergy: PCN and avoid all PCN & PCN-derivatives.
"A vaccine does not prevent infection, it primes the immune system to undergo an imeediate response to prevent an infection from spreading. Explain what that means." -- The statement relates to how vaccines work. I don't know how much you know so forgive me if I tell you what you already know.
The body manufactures many types of proteins that have various functions. Proteins that originate outside the body are regarded as foreign. Foreign proteins are called antigens. Sources of antigens include pollen, pet dander, donated organs & tissue, including blood, and bacteria & viruses. Bacteria have proteins on their surface. Viruses are surrounded by a capsule that has proteins on its surface. It doesn't matter whether the body is accidentally exposed to a bacteria or virus or if it's been deliberately introduced into the body, as with a vaccine, the body will respond in the same way. But with a vaccine-induced "infection" the extent of the "infection" will be necessarily limited. This may be because dead or inactivated organisms were used or because live/active attenuated (weakened) organisms were used. Sometimes all it takes is the antigen to do the job.
With a vaccine there is either no reproduction/replication or the reproduction/replication is limited. But the presence of the organisms is enough to stimulate T-cells into initiating the immune response and effector B-cells into producing antigen-specific antibodies. Memory B-cells "remember" the specific antibody to manufacture for the specific antigen that was encountered. Antibodies remain in circulating blood although the numbers drop off over time. If the same organism, same strain, is introduced at some future time, the memory cells recognize the antigens and antibody production immediately begins to bring the levels back up again.
So what I believe the statement means is that a vaccine doesn't prevent an organism from entering the body and starting reproduction/replication. This is what infection is. Vaccines prime the immune system by introducing it to a pathogen and thereby triggering a response. The memory B-cells will handle an immediate response to prevent an infection from spreading should the same pathogen be introduced again because making antibodies it's already produced before takes less time than making new ones. The B-cells already know how to make them. Plus, there will be some in circulating blood that can immediately go to the site of the infection.