Mental entities are just like other entities; they are made of smaller components.
But, says one, my perception of myself is that I'm a single entity, not made of fragments.
Well, of course. It couldn't be any other way. When you look at a brick, do you see the molecules that compose it, or the brick as a whole? There's two reasons that you don't see the molecules. The first is that molecules are too small, your vision can't see them at all without outside assistance. The second is that even if you could physically see them, your vision abstracts away all the detail and just leaves you with the impression of a "brick" unless you're explicitly looking for the details.
It's the same when you introspect (look at your own mind). First: you can try to figure out why you have the feelings, etc., that you do; but you can't introspect down to the neuronal level. You can't tell which neurons of yours are firing. To expect to be able to do so is like expecting to be able to see the molecules that compose a brick. And second: as one example, anger and happiness are already two different components of your mind. You probably have never thought about it like that before, because it doesn't feel like, "my anger module has activated," it feels like "I am angry" (and actually, by the time it feels like that, you've probably already been angry for a while). Cognitive science seems to be moving towards the view that the conscious you is sort of like the brain's press secretary: it is trying to make a coherent story out of your past actions, both for your benefit and that of your peers.
And now, if I may harp on this for a bit:
Just because anger is, in fact, electro-chemical patterns of activity spread across billions of neurons, does not mean it is not real! It is not "mere" electro-chemical patterns of activity spread across billions of neurons, it is also "anger"! Just like you still call a brick made of molecules a "brick", you should still call anger made of electro-chemical patterns of activity spread across billions of neurons "anger"! Happiness is likewise electro-chemical patterns of activity spread across billions of neurons, but it is still happiness. Love is electro-chemical patterns of activity spread across billions of neurons, but it is still love!
You see that we have explained (partially) emotions and minds, we have not explained them away. They still exist. They are still amazing.
(Disclaimer: "electro-chemical patterns of activity spread across billions of neurons" is a description of the process, not a complete explanation. If I acted like it was enough info to go out and implement the emotion in an artificial brain, I'd be committing a sin for rationalists, which is why I'm including this disclaimer. At the same token, just because we don't have a full explanation doesn't mean that what I've described is wrong, so you can't use this as an excuse to ignore this. :) )