That message caught me by surprise as well. I assumed they were heading in that direction when they demanded the use of API keys, but I didn’t know when or if they would require billing info.
There’s unfortunately no way to reduce the API calls – MapPress already uses the API as sparingly as possible. One suggestion: if you show maps in post excerpts you might consider limiting them to just single-post detail pages.
Google does provide a ‘user guide’ that explains how to estimate your future billing:
It looks like the first 28,000 maps are free. In months where your site gets heavy traffic and exceeds that free allowance, they show $7 per 1,000 map loads.