You move your piece to Jail and must wait until the next turn to play the card.
The official rules only talk about using "Get Out of Jail Free" in one place:
You get out of Jail by... (1) throwing doubles on any of your next
three turns; if you succeed in doing this you immediately move
forward the number of spaces shown by your doubles throw; even
though you had thrown doubles, you do not take another turn;
(2) using the “Get Out of Jail Free” card if you have it; (3) purchasing
the “Get Out of Jail Free” card from another player and playing it;
(4) paying a fine of $50 before you roll the dice on either of your next
There is nothing in the rules that suggests you can use this card to avoid going to Jail.