Out of the box the Refinement web part shows the different filters, but it doesn’t show the number of count for the results.
For this we need to modify Filter_Default.html page that is being used by these filters.
Go to the master page gallery
Select Display Template à Filters à Filter_Default.html
Download the file
Make the following change in it
Change the ShowCounts to true and upload the file.
Do the search again and we will see the count
Hope it helps.