You will not be obliged to "prove" your hours. The honor system is used. Keep track of your own time. The application will ask you to provide a contact person and the location of the experience. Many will also give a phone number or email address for the contact within the free-text box.
Clinical experience is an unwritten requirement at most schools. Rarely, someone might get an acceptance with none. It's your job, however, to develop an application that will appeal to a wide range of schools, so as to give yourself the best shot at getting an acceptance. I have seen many applicants told (post-rejection feedback) that their clinical experience was inadequate when they failed to gain an acceptance.
The average applicant, from my observation, lists about 1.5 years of clinical experience during which time they may accumulate 150 service hours. Note that the duration is more important than the hours per week, as adcomms like to see that you've thoughtfully tested medicine as a vocation over a period of time.
You can get clinical experience with sick people through the workplace, for class credit, data gathering for a clinical trial, or via volunteerism. It can be gained at a free, family-planning, or private clinic, hospice, hospital, VA, residential home, rehabilitation facility, nursing home, as a first responder, among others.
Clinical patient experience is not always gained in a clinical environment, eg EMT, battle field medic, home hospice care, physical therapy aide, special camp environments. In such a case, you also should also acquire some experience in a clinical milieu where doctors work.
The advantage of gaining clinical exposure through volunteerism, is that it also is looked on as community service, another (usually) unwritten requirement for your application.
However, we are increasingly hearing that adcomms express an interest in seeing nonmedical community service in addition, so maybe plan to give some time to a noncampus cause you care about, as well.