You skip the draw and discard right away.
On a normal turn, you draw one tile from the Draw Pile. Then you discard one tile. Taking a discard is the other way to get a tile. So you do not draw after taking a tile in Mahjong Island.
What happens when you take a tile?
You use the taken tile to make a Group, lay it face up, then discard. A Group here is 3 to 6 identical tiles that a hand on the Island Card still allows. The smallest is a Set, three matching tiles (called a pung in traditional mahjong).
Say you hold two 7 Dot tiles, and a hand on the card needs three. Another player discards a 7 Dot. You take it and lay all three face up as an Open Group, which every player can see. Then you discard one tile, with no draw first.
What if two players want the same tile?
A Win comes first. If nobody wins on the tile, the player closest after the discarder gets it. That player then discards, just like any other take.
What if the tile wins the hand?
Then you do not discard at all. You can Win by taking a discard that completes your hand, and the hand ends there.
To see every rule in one place, read the Mahjong Island rules guide. To learn how lines on the card are built, see how the Island Card works.