Thanks to Apple's configuration profile, you can download iOS betas directly on your iPhone or iPad, no Mac required!
Press Restart to apply the changes to your iPhone.
To install the iOS 12 beta, you'll need to visit Software Update on your iPhone or iPad.
Enter your Passcode.
Your iPhone or iPad will reboot to install iOS 12. There will be a couple progress bars going across the screen. Sit tight, and let everything finish.
Once your iPhone or iPad reboots, it'll be running iOS 12. There's a brief bit of setup you'll need to go through, however, as the system will have changed enough that it'll need your login to complete the update.
Once that's done, you'll be on the Home screen and ready to rock iOS 12.
A lot! iOS 12 comes with a number of new features, including new capabilities for ARKit, new features for Siri, and new device time management abilities that allow you to better control how you use your devices.