I've gotten maps from quests that increase a horse's traits by +10 or +20 but what does that mean? I know that breeding items like svens, medallions, and glass can also increase these.

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Is it the number of bars in their training gauge? Is it a hidden number that determines how fast each skill trains up?

It's a hidden number that affects their rating, not how fast they improve. The maps can only be used on wild horses that you capture, the breeding items are only used on horses you breed. Look at the first horse in your barn, Wild Wind Jeopardy, whose rating is EPP. Using some kind of a breeding item might have given her a rating of EEP. If a jumping medalion or a Scope stained glass horse had been used, her scope might have been equal to or better than the other stats around it, instead of being her weakest stat. Since the breeding items tend to be pretty expensive, they're usually used on high-level matches, like WWW x WWW. Using a Sven might boost the foal from an EWW to a WWW, which is a huge difference in the breeding world. The maps are also a gamble, since you can't tell what level of horse you're going to capture. The map might improve your horses ratings, but it might only be a sub-sub-sub to begin with, so you might use a map and still get only an AAA. Hope that helps!

Edit: to flesh out my answer a little more: Your horses have 6 "traits" each, starting with Movement (mvt), Intelligence, (int) etc. Each color on the training gauge is a training level, 2 bars per level. So looking at Wild Wind Jeopardy, I can see she's about to level up in Dressage, because her training level in Dressage is a 3 but she has 4 colors showing in the dressage traits of Int and Str (strength). I believe each bar takes 4 points; each time you train you get 6 points, 1 per trait if you're training all disciplines, so it takes 4 weeks of training to go up 1 bar (on average). The weaker a stat is, the longer it takes to go up. For instance, a WWW horse might show an increase every 3 weeks, and a PPP might increase every 5 to 6 weeks. Since she's an EPP, her dressage traits will go up faster than the others.