You're not keeping enough pressure on Link. There are times when you knocked him down, or had him locked down, and you suddenly just stood there and waited. The more pressure you keep on Link (by dashing in and out, approaching w/shffl'd n-airs, empty shorthops, etc.), the less time the Link has to set up projectile walls or space with sword swipes.
You're also relying on the knee too much. The knee is great and all, but in general, it's used mostly as a combo finisher. Very rarely do spammed knees actually connect; you need a decent setup--usually from d-throw/u-throws, n-airs, stomps, or side-b's.
Basic Falcon staples are absent. Dash-dancing was used improperly; you want to use it in such a way that your approaches and grab patterns become unpredictable. Shffl'd n-airs weren't executed correctly, and weren't used as often as they should have been. There wasn't as many grab combos as there should have been. Techchasing was absent.
Falcon is a very grab-centric character. In that sense, he's like Mario--once you grab, you basically get a free combo. D-throw at lower %s on Link can lead to techchasing opportunities, at mid %s can chaingrab into another d-throw or n-air juggling or knees. U-throw at low-mid %s leads to shffl'd u-air juggling. Please, please focus on grabs, because that's where the majority of his damage comes from--grab combos.
Learn to techchase, too. Techchasing is usually done after a d-throw (on Link, anyways); you want to guess if they'll techroll left, right, or in place, and aim a side-b/stomp at the place they'll end up. Once you get a handle on techchasing, it opens up a lot of grab/combo/knee setup opportunities.
Link is very, very easy to combo. Just get a grab, and you should be set. But you need to learn to pressure properly and space properly, because the Link was pinning you down with both projectiles and sword swipes.