Traditionally, treatment is 7 day course of penicillin.
However, people stop after 3 days so reinfection is common.
A new antibiotic needs only 3 days (to work fully)

Conclusion: Re-infection will be less common in case of new antibiotic.

This should be true if people will take the new antibiotic for 3 days. Option (E) tells us that this is what happens. Most people feel fully recovered after 3 days whichever treatment they take. So it is likely that with the new treatment reinfection cases will be fewer.

A. Some of the people who are allergic to penicillin are likely to be allergic to the new antibiotic.

These cases are irrelevant to our argument since they can't get either treatment anyway.

B. A course of treatment with the new antibiotic costs about the same as a course of treatment with either penicillin or erythromycin.

The comparative cost of the two treatments is irrelevant. Patients who are prescribed a certain medicine will need to take that medicine since without prescription, they won't get anything else. So people who are prescribed a medicine is the same set as the people who take that medicine. Note that the same language is used in the second sentence as well "... reinfection is common in cases where those drugs are prescribed". Cases where it is prescribed = Cases where the patients take it.
Patients don't have a choice, whatever may be the cost, high or low.

C. The new antibiotic has been shown to be effective in eradicating bacterial infections other than strep.

Other infections are irrelevant.

D. Some physicians have already begun to prescribe the new antibiotic instead of penicillin or erythromycin for the treatment of some strep infections.

Irrelevant.

Answer (E)