Hume is purposefully exaggerating to make his point about the sentiments being the ultimate reason that we do anything. He does not mean that reason is incapable of leading to action. He does mean that reason alone is incapable of leading to action. So it’s not that reason and emotion are opposed to one another, but rather that reason relies on emotion.
Think about it this way: reason is really good for doing means-end reasoning, but where do you get the ends? If I tell you that you need to do A to get result B and that you need to do C to get result D, you can’t figure out whether you should do A or C until you figure out whether you want result B or result D.
To put this in more practical terms, let’s say I offer you the contents of Box A or Box B. Presumably, you’ll want to know what’s in them before choosing. That’s because the logical choice is to pick the one containing whatever you want more. But wanting is a feeling. It is the input your reasoning needs to figure out what sort of output is most appropriate (that is, choosing Box A or Box B).
The same thing, according to Hume, applies to ethics. When you watch someone do something terrible, you can’t find the badness of their action in any scientific description of their action. The description of a blade cutting open skin is the same in the case of a murder as it is in the case of a life-saving surgery. How we feel about the various cases—about the motivations for the actions and their consequences—is important in figuring out whether they are right and wrong.
This isn’t to say that the matter is wildly subjective, however. How we feel is grounded in human nature (which is more or less universal) and tempered by what Hume calls “the general point of view” (because we have to abstract from our own biases to judge a case fairly). Nevertheless, reason can never make a judgment without the input of reason. And so, reason is the slave of the passions (it’s just the way things are) and ought to be the slave of the passions (giving it more of a role would lead us astray).
Interesting note: a lot of contemporary work in psychology and neuroscience suggests that Hume is completely right with regard to his claims about human nature and how moral and practical reasoning work.