It's pretty simple, really. You probably already know most of it, I just think it's really important in some matchups (it helps a lot in the ditto, and D3 from what experience I have against both characters, but tbh I don't have much experience vs most characters anyways). Easy example would be in the D3 matchup.
Say that D3 is in the middle of FD vs MK. MKs back is to the ledge. D3 can literally go WHEREVER the **** he wants, and be defensive, and retreat a lot. MK loses his ability to simply walk around, retreat, etc or else he'll go off stage. MK really isn't that good offstage when his opponent is on stage because his options are still limited a lot, and he's far more predictable. MK is only good on the ledge when he doesn't need to get back onto the stage lol (IE planking or scrooging).
If you want to take the safe option and retreat, you're now off stage. You now, because of your low mobility, have no good way to crossup to get back stage control, and your options are limited to like all aerial stuff. Your getup attack, which has the most range on the ledge, is slower than reaction (by slower than reaction, I mean it's slower than reaction in the practical sense. I have a slow *** reaction time, and it's like 3 or 4 frames slow than that. It is DEFINITELY shieldable on reaction). You just don't have very many options, and he has like every option available to his character, and mostly all of it is safe if he stays at a good position. Your ways of getting back onto the stage are risky, and don't have much reward at all apart from no longer having **** for options (which is a good reward and all, but... well you know, it's just not as much potential reward as MK usually has from every other position in this matchup).
So basically, when your back is close to the ledge, your risk:reward changes a lot. Suddenly you have less options, your opponent has more, and the risk for getting hit is higher because if you get hit you suddenly have WAY ****tier options.
Stage control is also based on what direction your opponent is face and stuff like that. Like if your opponent has the center of the stage, but is close to you and is facing away, and doesn't really have many options for hitting you (say they're in shield), you probably have a slight position advantage; because he's limited in what he can do to you. But generally, being closer to the center of the stage is better.
Now see what it's like when you're in the center of the stage, and D3 has his back to the ledge. What risks are there for the D3 if you get a hit? He's offstage against METAKNIGHT, with a recovery that gets gimped REALLY hard, and ledge options that aren't safe, nor very reliable (that's not to say that I'm theorycrafting that D3 can't get back onto the stage or anything like that, his options for doing so in that position are just REALLY ****ty, and a lot of them can be covered on reaction).
If D3 gets a hit, either he has MK off stage, at a high angle (from landing a Bthrow), which isn't THAT bad of a position for MK. At a high angle he can get back pretty easily through gliding, dair camping, tornado, or if you're not playing on FD (and you probably won't play on FD in a super srs tourney match, but playing on exclusively FD for a while helped my stage control a lot, and learning my options from the neutral position, and makes it easier to present an example), land on a platform where you're, again, not in TOO bad of a position (a disadvantaged position if you're supposed to be on the offensive, but seriously not that bad). Or he'll hit MK with something that knocks him forward towards the center of the stage again, and all MK loses is some percentage, and the position being reset to mostly neutral (depending on a ton of factors like percentage, but still mostly neutral if you get hit by most of D3s moveset at the position mentioned earlier).
So we know that the risk:reward ratio is very much in your favor when you have the center of the stage, since most of your attacks put him in a like the worst position in the game.
But also note that D3 has less options when his back is to the ledge, and it becomes easier to bait stuff when their back is to the ledge. Mango in Melee is a ****ing genius at this. I mean he generally just reads people when they roll, but a lot of people roll specifically in this position. Any of you watch Mango vs Shroomed?
http://www.youtube.com/watch?v=bSBa-tSv7ts
But it's not just rolls, people get really predictable when they're pressured because it's a really uncomfortable feeling to lose a lot of options. D3 CANNOT retreat when he's in that position, which is usually one of the safest options in the game, whereas you can (hell you see Mango do it in that video (make sure you listen to it with sound on), when he's waiting for Shroomed's roll, and he doesn't roll, so Mango just resorts to getting to a safer, but still advantaged position).
Melee and Brawl are really different games, and stage control kinda seems to work different in Melee, due to the completely different risk:reward scenarios from the more powerful gimping, increased movement options and all that. But that match is still a really good match to watch.
And they are closer to the blast zone than usual so they die earlier.
They have less options, you gain more options, they become easier to bait, you can be less predictable (due to your increased options), you reward for landing a hit (especially a grab) becomes HUGE, your risk for getting hit becomes smaller because you'll likely still retain all of the above bonuses if you're hit from this position at a low percent.
It's just REALLY helpful, and it's easy to do, and something that you can begin to do without using focus if you practice enough. And from my experience, people who don't get stage control all that well are REALLY easy to bait and punish because they don't even realize when they're being pressured, often, or that their opponent is waiting to see what habits they have when under pressure.
Ledge pressure is also REALLY helpful. The better your ledge pressure and stage control are, the better your followup game is. Followups aren't necessarily about getting like 10 straight hits, but also about getting hits that put you in a far better position than your opponent so that you'll likely continue hitting him. And followup game is obviously very helpful.
And knowing when to chase your opponent offstage, and when to just wait for them onstage (staying on the ledge while they're off stage is a good way to keep both the chasing after the opponent option, and the retreating onto the stage option open at the same time, imo).
Although I'm bad so I could be completely wrong on all of this, just my opinions on stage control
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