This is the number of new contacts that have been added to the HubSpot CRM (sorted by referring site) during the time period for this report. You can change the time period of your report by using one of the dropdown menus above the table.
You click on the number in the contacts column to drill down to see the details of each new contact. This is a great way to see the general quality of sales leads you're getting.
While in the Noisy Little Monkey example above there are no conversions of contacts to customers, HubSpot will collect this data for you so you can see which of the referring sites are generating leads that eventually turn into customers.