For a really good answer, please see this site. My answer comes from it as well as the answer to this question. Transitive and intransitive verbs can be hard so research is always the best course of action.
Transitive verbs take someone to do then. They do not happen on their own. There is a subject doing the action, and an object that receives it. Here are some examples:
I broke my phone.
I moved the car.
Intransitive verbs do not have an object. There is, of course a subject, but these verbs happen on their own since there is not an object. For example:
The car moved.
My phone broke.
So why is 遊ぶ intransitive? You do that on your own. To play, you don't necessarily need an object. Sure, you can play with the ball, but play is still intransitive here. If we were to use play as a transitive verb, it would be:
I play the ball.
That is a transitive usage of play, and it doesn't make sense in English. In Japanese, there is no transitive counterpart for 遊ぶ because it wouldn't make sense even if it did exist.
Just as a side note, playing an instrument (like the piano) is a different word in Japanese, ひく。