Late afternoon downpours are the order of the day in Central Florida during August. We were there in August last year and it rained nearly every day like clockwork between 2 and 4 in the afternoon. The only days it rained a LOT were when the remains of tropical storm something-or-other went through.
Pack ponchos, and keep on touring. Rides that are predominantly outdoors (BTMRR, Primeval Whirl, Kali River Rapids) should be done in the morning when it hardly ever rains. Kilamanjaro Safaris will continue to run, but you may not see many animals. Try to plan indoor rides and shows for afternoon if you're going to be in the parks that time of day. Don't forget to use FastPass though since most of those rides will be busier in the afternoon. Also, Haunted Mansion, Pirates, Peter Pan, and others have mostly indoor waiting areas. And of course, you can always plan to eat a late lunch during the rain - but the restaurants will fill quickly when the drops start falling.
All in all the rain didn't really impact our enjoyment of the parks. The only thing we never got to do was Primeval Whirl (a favorite of ours) due to lightning from the tropical storm leftovers.
Have a great trip!