Many applicants have friends or family that have already matched and are currently working as residents or others that are in the medical profession. Some of these applicants will reach out to their friends or relatives to put in a good word with the program coordinator or program director in an attempt to help their application.
References are a legitimate way for managers in corporate America to identify good candidates for open positions. An employee vouching for a colleague from a past job or from a personality standpoint is helpful to managers. In the context of the residency application process a candidate may receive an interview invite on the basis of a reference however I believe only in certain cases will that matter much.
If a program director or a selection committee does not send out an invite to an applicant its probably based on particular criteria they have set which the applicant does not meet – in this case enlisting the help of a friend or relative to gain an interview might not mean much in the long run. On the other hand being acquainted with a faculty member or program director that is aware of your credentials/character might be more beneficial.
In cases where the candidate may have an exceptional personal story which would not otherwise be apparent on an application it might help. It might help during the ranking process when it comes down to ranking two identical applicants, having someone personally vouch for one of them might provide an edge.
Remember your ultimate goal should not be to rack up interview invites but rather to be ranked high by the programs where you interview. In my opinion interviews gained based on merit or strength of an applicants profile have much stronger legs to stand on