"Overdrive" originally came from cars, as a way to make the engine go faster.
The phrase "Kick into overdrive" means to do something extra vigorously - go faster, work harder, etc. In this case "kick" means to starting to do something very quickly - like kicking the ball makes the ball go faster, it can refer to starting any action immediately. Also, see "Kick in the pants." - another popular saying in (American) English.
"The student's motivation to get an "A" kicked his studying into overdrive."
In terms of health, when your body uses calories to make energy, we say it "burns calories" - like how a furnace or engine burns fuel.
So the whole phrase "kick your metabolism into overdrive and burn body fat" means this is something that will quickly cause your metabolism to increase (or speed up), so your body uses your body fat - and you lose weight.