There is really not a "rainy season" on Ambergris Caye in the sense that many people think of tropical or sub-tropical rainy seasons. Yes, beginning usually in late June the very welcome seasonal rains come to Belize, but it's not as if there are monsoons. In any given week in the summer and fall there can be some days with a lot of rain, especially so if there's a tropical storm in the region, but again it's like many places in the world including much of the U.S. -- you can't predict the weather on a daily basis. It could be dry and sunny for days on end, or you could hit a rainy spell. Most likely, even in the so-called rainy season, on the cayes it will be sunny part of the time, partly sunny or cloudy part of the time, sometimes with showers or rain for a couple of hours and then clearing.
Overall, I agree that June is a great time to visit the cayes. It's before the prime hurricane season and after the prime tourist season.