A little more context would help, but "push 'play'" means to push the 'play' button (the triangle pointing to the right) on an electronic device (usually a DVD player, CD player, or video game). However, that phrase could be used as a short-cut for "turn it on" for something that doesn't actually have a "play" button.

This would suggest a DVD, cd or similar. The plug it into something, would be to put it into a suitable player and "push play" would mean to start it playing straight away. The push is from when video recorders were used and they had physical buttons, and the play is to start the programme, song or whatever is on the object. I hope that this makes sense

below (↓) ǂ above (↑) you can't say " push play " , but correctly you say " press play " which means to press the butten that makes ( the game or the video ) goes-on , starts , continues playing ....