Before you can take advantage of third-party filters in Photos, you'll need at least one app installed that supports Photos extensions. I have Camera+, Paper Camera, and Readdle's Scanner Pro installed now, and I'm impatiently waiting for Waterlogue to get updated. Ironically, Waterlogue was the app Apple used to demonstrate iOS 8 photo filter extensions at WWDC earlier this year, yet an iOS 8 Extensions-savvy version hasn't been released.
Photos without any Extensions available (left) and with Extensions (right)
To add an app's filters to Photos, make sure to launch the app at least once so you can give it permission to access the photos on your iPhone or iPad. That also lets Photos know new third-party filters are available.
Once Photos knows your Extensions-ready apps are available it'll show a new icon along with the crop, filter, and adjustments icons in the Edit view. The new icon looks like a circle with three dots inside, and tapping it shows which app's filters you've enabled.
No third party filters showing up (left)? Tap More to add some (right).
If you haven't enabled any filters yet, just tap the More button, then enable the apps you want. You can also reorder the apps from here. Tap Done, and then tap the app with the filters you want to use. Enabling third-party filters is a one-time process, so you won't need to worry about going through the setup every time you want to edit an image.
Tap an app (left) to use its filters in Photos (right)
Tapping an app icon in Photos shows its filters and lets you use them as if you were in the third-party app instead. Once you're happy with the edits you made, tap Done, and your image updates just as if the changes were made with filters built into Photos.
Why bother enabling third-party Extensions in Photos, you ask? While you could jump from app to app making edits and changes, having a unified interface is handy, and you don't have to deal with duplicate images if you need to use filters from multiple apps: one app means a single image for all your edits.