What happens in Exploding Kittens when the only card in your hand is a skip card? Do you lay it down and still pick up a card anyway?
On your turn you can play any number of cards from your hand, including zero cards. So you only need to play the Skip card if you think it is worth it.
If you did decide to play the Skip card, your turn ends immediately without drawing a card.
If you choose to not play the Skip card, you would just draw a card.
From the rules (page 1 under Taking Your Turn):
1 Look at the cards in your hand. Play a card by placing it FACE UP on top of the Discard Pile, and following the instructions on the card.
2 After you follow the instructions on the card, you can play more cards. You can play as many cards as you’d like.
Or, play no cards at all; that’s cool too.
And on page 2:
Immediately end your turn without drawing a card.
Note that the linked rules are for the Party Pack Edition, but the base game rules say essentially the same thing in a slightly different way.