There is (currently) no easy way to mass select or mass delete your captures on Gyazo. If you would like to delete all your captures without much hassle, try this script. (Last tested on 1st of May 2020)
Copy paste the contents of gyazodel.min.js in a chrome developer console on https://gyazo.com/captures.
2 buttons will appear in the navbar.
Click the "Delete All" button to execute an automatic mass deleter for all visible items (this will take about 5 seconds to complete).