This is a unified “How It Works” section for all incarnum-using classes. That’s Incarnate, Soulborn, and Totemist.
Incarnum comes from Magic of Incarnum, one of the later entries into the 3.5 library and one with fairly little support. Dragon Magic has a few more soulmelds, as do web supplements. I don’t know if Dragon Magazine ever put anything out, but really Magic of Incarnum has nearly all the incarnum material.
So, to start off, what is incarnum? It’s essentially cosmic soul stuff. Incarnum is the energy of all souls, whether they’re living, dead, or not-yet-born. Any mana analogue works, though.
Also, incarnum is blue.
There are a few vocabulary words to know when dealing with incarnum, and really, they’re the hardest part.
Soulmelds: These are the bread and butter of incarnum. Basically, a soulmeld is where you take a bit of incarnum and shaping it into a tool and melding it to your soul (hence the name) for a time. You can think of a soulmeld as a kind of phantom magic item, or a visible buff. Each is associated with a part of the body (those Cerulean Sandals go on your feet), and tend to manifest as ghostly body parts or equipment. They each have a certain benefit, like bonuses to skills, AC, or AB, or more interesting abilities like the ability to spit acid or fly short distances or inflict fire damage on foes who strike you in melee. These benefits improve via the next two vocabulary words. Soulmelds do not have levels, and all meldshaping classes gain access to all their soulmelds automatically. The number of soulmelds you can shape at a time are limited by your class level and your Constitution score.
Essentia: Personal soul energy. Mechanically, essentia’s just a pool of points that you shuffle around your soulmelds (or in some cases, feats and abilities) to improve your benefits. So, the Gorgon Mask increases your fortitude saves by one, and for every point of essentia you have invested in it, the bonus improves by one more. Essentia can be shuffled between your soulmelds as a free action once per round, letting you adapt to changing circumstances. So, when you really need to strangle somebody, you can shunt that essentia from your Gorgon Mask, losing the fortitude boost, to your Girallon Arms, gaining a bonus to grapple checks. Essentia can be gained from race or magic items, but it’s mainly a class feature. The maximum amount of essentia you can invest in a single place is limited by level, going from one at level one to four at level eighteen. Certain feats and class abilities can improve this, up to about seven or so.
Chakras: Every character already uses chakras. They just go by a different name; magic item slots. Normally, soulmelds don’t use item slots, so you can have a soulmeld associated with your throat and still use a magic amulet. However, as your level goes up, you open up chakras, letting you bind soulmelds to them. When you bind a soulmeld to a chakra, it becomes more powerful but uses up a magic item slot. So, for example, Mauling Gauntlets normally grants a bonus to strength checks that scales as you invest essentia. If you bind it to your hands chakra, it takes up a magic item slot but grants you an equivalent bonus to unarmed damage along with the benefits of Improved Unarmed Strike. Which chakras you can bind to and how many soulmelds you can bind are limited by class level.
And, with that vocabulary, that’s pretty much the bulk of it. Just get your eight hours’ rest, do your hour’s spell preparation- in this case, it’s shaping and binding soulmelds for the day- and go about your adventuring way shuffling around your essentia between your soulmeleds as needed. It’s a very simple system.
Characters who don’t take any levels in incarnum classes can still get in on the incarnum game. Shape Soulmeld lets you shape a single soulmeld, and it’s an awesome feat for filling in blanks if you ever have a feat to spare. Almost anyone can put it to use. The Azurin race are basically incarnum-infused humans who get a point of essentia instead of bonus skill points, and an Essentia Jewel is a cheap magic item that can get you a second point. So, if you’re planning a human Paladin of Kossuth (FR’s god of fire), you can go with an Azurin and spend a feat and a few gold to get Shape Soulmeld: Mantle of Flame so that anyone who hits you in melee takes 3d6 damage (2d6 before you hit level 6).
Incarnum has a lot going for it, but ultimately, it has one major flaw. It doesn’t really scale well. Essentia capacity goes up with level regardless of whether you take incarnum classes (though incarnum classes get a little more capacity than non). You get all your soulmelds at level one (or level four for Soulborn, but they suck). All you really get for leveling up is more essentia (which is admittedly very nice) and the higher-end chakra binds, which tend to be underwhelming for the number of levels you have to invest to get them.
On the other hand, incarnum makes for some great dips. 1-4 levels in Incarnate or Totemist get you a huge stack of soulmelds, some essentia to play with, and your basic chakra binds, and you can just multiclass out to some melee class or skill monkey or something with a nice sack of goodies. There are even a great many incarnum feats that are explicitly designed for multiclass incarnum-users, like Cobalt Rage (which you can invest essentia in for bonus damage while raging) or Indigo Strike (which you can invest essentia in for bonus damage on sneak attacks). And while the incarnum dip is not a bad thing- in fact, it's a rather good and interesting thing- the fact that it’s so rough going straight meldshaper is a bit problematic (though it is entirely doable if you know what you're doing).
In addition to multiclassing spectacularly well, incarnum also fares well in gestalt, since it compliments everything. Not much more to say, there.