In iOS, open the App Store app. Go to the "Updates" tab (bottom right), and if Instagram appears under "Pending," update the app.
Troubleshooting note: Even after updating, I didn't see the GIF stickers update. I had to uninstall and reinstall the app to force the new feature to appear. If you don't see the GIF update at first, try that! The feature is, indeed, rolling out to everyone today.
In Android, open the Play Store app. Go to the menu (top left), select "My apps & games," and under the "Updates" tab, check for Instagram and tap Update.