This sounds dead simple, but having cleaned thousands of strainers, I’ve learn a few shortcuts. The following assumes we are cleaning a basket strainer with the debris on the inside. If the debris is on the outside reverse the process.

1. Scrape the outside with the back of a knife to shear stapled fibers. Alternatively, scrubbing with a wire brush may be easier, although the shearing action is not as strong.
2. Wipe the inside of the basket with a small brush in a smooth rotary manner, not back and forth. You want to pull the fibers out, not scrub them in.
3. Rinse from the outside with a strong stream. This should help push fibers through to the inside.
4. Repeat as needed.

Additional notes

A few small details can make a big difference when selecting the right strainer for the job

• Install valves near both sides of any inline strainer. If you can’t easily clean it, you won’t.
• Threaded covers are notoriously prone to jamming when over tightened. Clean the threads carefully , use silicone grease, and tighten only the bare minimum. Replace O-rings sooner than needed to avoid the need to over tighten. Get the right wrench, and only use it for removal (hand tighten only). Remove at least twice each year. 
• Plan ahead. How will you clean up the inevitable mess? There should be space for a pan.