By having the patient record all the places they went during the infectious period, medical workers can map all points of exposure from that source, allowing those areas to be given information and resources to suppress the spread of the disease. By recording public places visited by the patient, medical workers also gain an indication of possible exposure to persons unknown to the patient who are not listed on the contact tracing form.
By recording who was there at the same time as the patient, medical workers are able to quickly contact persons who have been exposed and who may be in infected without yet knowing it. This is vital -- by screening infected persons while they're still in the incubation stage for the full 21 days Rosling describes, the spread of the disease is completely stopped at that source. This also allows people who are infected to receive the treatment they need.