To add to Frederik's answer:
Jeg tror på dig! = You can do it! or I believe in your existence! or I trust you! or I believe what you're telling me!
Jeg tror dig = I believe what you're telling me
As Frederik said, you can use both in this example.
In this example you can say both. But 'tro på' can also mean to 'believe in (the existence or capacity of) something/someone', while if you just say 'tro' it can only mean believe if something is true or not.
So 'tro på mig' can mean both 'believe in me' and believe me', while 'tro mig' only can mean 'believe me'