Hi Mike,
I checked your app to see where the API activity is coming from. I see there are more than a few API requests. For example, on June 3rd alone, there were more than 4000 requests:
We pulled the access log from that day, and the requests are spread out throughout the entire day, starting from 13:31:11 GMT0 till 22:26:32 GMT0. Similar activity is occurring on other days (the first spike in the chart was on May 9th, with more than 10,000 requests).
Regarding debug sessions, they last 30 minutes and are indeed designed to allow for frictionless testing and development. When a page in testing mode opens, the system assigns a cookie that we use to designate the session as “in development.” Perhaps the browser you use rejects the cookie, or the cookies are disabled. This would explain why the API calls are counted. However, by looking at the activity log, it must be some wild development sessions 
To minimize the billing for the file-related API requests, I recommend integrating a CDN: Cache and CDN Integration - Backendless REST API Documentation
Regards,
Mark