You will receive a notification on Twitch. Make sure to claim the Drop on Twitch to receive it in the game. All rewards must be claimed before the end of the campaign (indicated, for example, by the countdown on this page).
Once claimed, you’ll receive an in-game message with the reward in one of the in-game chats.
Disclaimer: Please note that you will not receive in-game messages with Twitch Drops rewards while playing on a Custom Server. You’ll receive the message after you leave the Custom Server.