There are 2 ways in which SpeedyCache serves webp images:-
htaccess Rewrite Rules
To enable this you just need to enable Cache System as shown in the screenshot below and htaccess of your wesbite will get updated.
Rewriting Image URL
Rewriting Image URL simply updates the URL of the image in the <img> tag in html to the webp version if the webp version of the image is available.
To Enable Rewriting Image URL in your WordPress Admin go to > SpeedyCache > Image Optimization > Enable Rewrite Image URL as show below in the Screenshot.
If you face any issue do let us know and we will help you out, you can contact out support team at firstname.lastname@example.org