Imagine dwarf Alice (200 pounds) and gnome Bob (40 pounds). Both characters have a Strength score of 10.
So, as the PHB says, both of them are able to carry 10 × 15 = 150 pounds.
Does it mean that Bob’s effective carrying capacity is 150 - 40 = 110 pounds and Alice’s effective carrying capacity is 150 - 200 = -50, which means she has problems with carrying her own body?
Or does the character's weight not count for the purpose of calculating their carrying capacity?
No, carrying capacity applies to objects actually picked up and carried in the literal sense (which could include clothing and armor and things worn) but not one's self.
It's hard (perhaps impossible) to find a direct citation for this, but the carrying capacity rules are written in the plain English sense. However, there is an indirect citation:
From the PHB, emphasis mine:
Carrying Capacity. Your carrying capacity is your Strength score multiplied by 15. This is the weight (in pounds) that you can carry, which is high enough that most characters don't usually have to worry about it.
But, the random rules for rolling up height and weight (long enough I don't want to reproduce them, but reproduced and discussed in detail here) tend to put several races near or above their own carrying capacities with strictly average rolls, which means they definitely would be worrying about it.
It's just not possible to reconcile both of those rules, so falling back on the plain English meaning of carry-- pick a separate thing up off the floor and move around with it-- is correct.
No, a character’s own body weight does not encumber them. The word “carry” refers to other objects, not yourself. (“To carry yourself” is a phrase in English, but it has little to do with body weight in most cases, and in any event is not used by the Player’s Handbook.)
It may encumber other characters, if those characters attempt to carry them. This is most relevant when considering mounts. Most mounts are more than strong enough for this purpose, but in edge cases it can matter.
