(My roomie plays Falco so I play against him every day.)
I started writing this and got carried away so it's got stuff to help you but it also has stuff I saw you were doing ok.
Neutral Game
You may have noticed Falco is really slow horizontally. Falco can only approach with lasers, s-hops, platform fall through airs.
You'll see them try to approach with f-jumps, especially inexperienced ones, but he's in the air so long.
Responding to Lasering
When he's lasering you can f-jump or s-hop over (depending on laser height) and air or just keep dashing forward every time you get hit. It's actually a pretty bad idea to shield lasers, since shield limits your movement much more than lasers do! Falcos who are predicting you'll jump will shoot lasers just above your head to stop your aerial advance. Keep all your mobility options in mind when trying to get in on a Falco. More things you can do include maneuvering across platforms towards the Falco. Someone else said that FD is better for Falco than Fox... but because Fox can CG and Falco can't (if you DI away d-throw), and Falco can't extend his combos with ptatforms, I think the stage is stronger for Fox. It is harder to deal with lasering, but I think the CGs and lack of platforms for Falcos combos counter act that and more as long as you can respond to lasering properly.
Falco's Vertical Movement Restrictions
Remember Falco's jump height is shorter than yours when he's s-hopping, and the full hop leaves him in the air for a long long time. If you go high when he's lasering, he'll either just do a full jump or break up his ascent with wavelands. Either way you have a break from getting lasered in this time. Use this time to change altitudes again, but shake it up between top platform and going back down. Obviously you can also try staying at the same altitude as a further shake up, but some Falcos will just be shooting lasers there whether you're there or not so it's really not worth it to stay there until you know your opponent a little more.
Ground Control
If you're in neutral on the ground and he's not lasering you want to approach with f-jump nairs or fairs(approaching with fairs is weird so experiment/be careful), shines, and grabs. Also, dtilt is underrated at certain percents in this matchup, but it can pop Falco up for a tech chase or a follow up air. To me it seemed like you neglected to approach with airs enough, though ground control is really important for spacing, because Falco's s-hop is lower than yours, you can get away with nairing more. Mix up nairing at the beginning of your jump to shine and nairing late so you don't get anti-aired with nair. Be careful though because some falcos are pretty good at anti-air shines/utilts and they can start long combos off of that. That's when you want to empty jump waveland back or just shield. Run up shield is ok but unsafe, because shine grab is super strong with Falco so it isn't as useful as it is in other(floaty) matchups.
Baiting some Falcos across the stage is harder than others. Some are very aggressive and will be easier to set-up gimps on, and some will be content with hanging back and lasering.
On-stage
Shine
If you're not that close to the ledge, you can still shine him at higher percents when the hit-stun will be longer. Too low percent and he'll just grab ledge before you can get there. Try to combo or hit him with something stronger instead. If you're close enough to the ledge, you can turn around in the shine and grab ledge. If the Falco tries to just dj you can get another shine, or bair. Bair is safer at higher percents. But at low percents it won't work that well.
Getting a B-Throw
If they're coming after you getting a b-throw is simple. Just expect laser -> dair/nair, There isn't much else Falco can approach with because of his low horizontal speed. Be careful about your spacing under/around the platforms by the edge of the stage so you don't get hit. Stay mobile and don't throw out random moves if you're planning on getting the b-throw gimp. But if it's obvious they're likely to just hit you. Obviously this can't be your whole game plan. It helps to look like you're afraid.
If they aren't, you're going to need to combo for positioning, there are two good places to put Falco for the b-throw. Just try and get them on a platform above you near the edge, many Falcos try and d-air through the platform. Their only other option is running off the platform towards center stage or taking the top platform. (I know my roommate often tries taking center stage so he can laser me so I often just put the tip of my bair at the edge of the platform. If you notice the Falco is laser happy this read might work on them. Since this is the best movement path to get lasers on their foxy enemy.) The other thing you can do is just hit them near the ledge on the ground level. Then you can grab u-throw and you're in the 'Falco's on the platform' scenario again, or you can try and get in between them and the ledge, but this is really not safe. Falco's shine is so fast and combos well when you're under platforms so you're putting yourself in big trouble trying to get the gimp that way.
Using d-air or fair off-stage are going to kill you every time. But you can d-air the edge of the platform when Falco is up-b'ing back. Run off and shine turnaround b-air works well if you can read where they're going to be. Otherwise that's actually not very safe.
Covering Side-B
First thing you should be doing is watching for Falco to air control away from the stage, double jump, and then side b right back to the ledge. If he side-bs anywhere else you can get there quickly and punish with u-throw, b-throw, bair, d-tilt -> shine-turnaround b-air, f-smash, d-smash. only u-smash if he's high percents, unless you think you can juggle or get a tech-chase off of it.
If you see him air control away when he's been hit, you should grab the ledge. Falco's recovery limits him to side-b to the ledge, side-b high, and up-b back. All of these can be covered from the ledge. If he is up-b'ing back low, fall down and shine him, or time a roll up if he is low enough(Don't be wrong on this).
Shine Stall
Both Spacies don't utilize this enough. If you edge-gaurd on auto-pilot this technique will allow them back for free. React to what he's doing, don't just try and hit him with a b-air. Shine stalls are over quickly and then just follow the 'Covering Side-B' bit. Don't be stuck in any animations during the shine stall, or you'll miss your edge-gaurd. If they end up going low with the shine stall, then just cover the up-b back, which will likely end up as a roll up since Falco's up-b is really short.
Watch for the Meteor
When Falco is coming back he'll be able to hit you with the meteor. His side-b is almost as long as yours, and he goes through it a bit faster so he's harder to hit out of it. Your number one priority is not getting hit by it. If you hear the sound, it's basically too late, so watch for Falco trying to line up with you horizontally. Obviously if you do get hit by it try to meteor cancel. The timing is really tight and I'm not really good at doing it with Falco's side-b because I'm always surprised if it hits. ...You could try shine-stall, and when the time is right jc shine him out of that but there are probably better things to do like just grab ledge or b-air... even n-air.
Basically, you're going to want to hold the control stick down and away except when you're SDI'ing. If you aren't going to SDI, ASDI with the C stick away. Be careful to watch for what animations he's in and don't try and tech if he's going to laser reset you or hit you with something. If he is getting reckless with reads, condition him with rolls then tech in place and hit him with something.
Recovering
Don't burn your DJ to get on a horizontal plane to recover across, he'll just laser you into a position where he can drill you. If you make that mistake, get low enough where you can crawl up the side of the stage with your up-b(really hard on Battlefield, you can do it but it's very tight and for a very short time) and hold in and down on the c-stick and smash up and in on the control stick (to add the vectors. If you do it in the same direction the c-stick one will just out prioritize the SDI and the SDI one is bigger so it's better. Then just hold up and tech the d-air and side-b back(up-b if he went real low to hit you and the stage has a wall there).
Also don't forget to recover high, it's really good.
If you've grabbed the ledge, you can tournament winner(don't!), ledge-dash, roll up, stand up, ledge-hop air, or ledge hop side-b(this is also usually not a good idea)
If you over use any of these, you're going to get punished. Falco can cover ledge hop, roll up and stand up very well. Ledge dash (especially invincible) is extremely good in this match-up because your horizontal speed is so high. Falco has to just hit you or jump to flee. If you're invincible then you get a free hit if he tries to just hit you. otherwise you're in a situation where the Falco is above you, which means you have the advantage. He could also back up and laser, in which case you take the edge platform and come down on him for lasering, or threaten to and take top platform or go back down to the ground if you think he'll go up to laser you.
