Correct spelling, explanation: this form is correct because it is the third person singular of the verb play. To create such a form we add -s to the end of the word. In this case, the word ends with y preceded by a vowel, so we do not change nor add anything else; we simply add -s, so we receive the word plays. Playes is incorrect because there is no need to add also e in between. Plays is the only correct form.

Definition of plays:
verb, third form singular of the verb play, to act (in a play, a film, etc.); to do an activity for pleasure or sports
Sarah plays chess but Mary doesn’t.
Everybody in this film plays incredibly.

Incorrect spelling, explanation: this form is incorrect because there is no need to add e before adding s. Plays is the third-person singular form of play, and we simply add s to the word to create such form. Therefore playes is incorrect and plays is the correct form.