Hi Tobias.
The translation is wrong because, as you noticed, it's not a past sentence.
Also, it's difficult to translate because あそびます doesn't translate exactly; it can mean 'hang out', 'play', etc.
...AND they are using たのしくas an adverb,..
So it means something like: "I enjoyably play/hang out with friends".. in normal English:
"I have a nice time playing/hanging out with friends",..
.. I would say. :)