Trusted devices let you skip the two-factor part of logging in. If you try to access your Riot account on something that you haven't designated as a trusted device, you'll have to check your email for a code to input before you can log in. This gives you all the convenience of a normal login with the added security of two-factor authentication.
Checking "Add this to my trusted devices" upon login will prevent future verification requirements on that device. Here's what it looks like:
Managing trusted devices
Visit your account settings to clear all trusted devices on your account. Future logins from all devices will require a 2FA code, but you'll be able to trust them again as you log in. This won't log you out of any logged-in devices.
Unfamiliar trusted devices
You'll get an email when a new device is flagged as a trusted device. If you didn't add it, or if you don't recognize the device information, clear your trusted devices immediately. If you use that password elsewhere, reset them all.
How we recognize trusted devices
When you log in, we check your cookies to see which device you're using. Trusting a device lets us store those cookies so we know it's the same next time. These cookies don't contain any personally-identifying information, just enough to make logging in a little more convenient.