Florida resident here. In June, you can expect muggy hot days, afternoon thunderstorms, and humid evenings. The specific weather will depend on where you are. We live on the coast. Coastal communities are significantly cooler than inland areas. Air blows over the cooler water of the ocean towards the warm air over the land creating a sea breeze. At night, the opposite happens. The cool air blows from the land towards the ocean. Either way, we get the benefit of cooling winds. With all the mixing around of hot and cooler air and very warm temperatures, we get afternoon thunderstorms. They tend to cause very heavy rain and lots of lightening, but they are gone as quickly as they arrive.
A typical June day would bring highs in the upper 80s to lower 90s, very high humidity, and afternoon/evening thunderstorms. It's not a bad time to visit. In Florida, if you don't like the weather, just wait a few hours. It will change.
I should add that June is the beginning of our hurricane season, so make sure you watch the weather.