There’s been some good answers in this thread—yes, philosophy does help us sharpen our cognitive faculties; and yes, philosophy gives rise to the individual sciences (actually, I don’t believe there’s really a useful distinction between ‘science’ and ‘philosophy’—it’s just that people seem to take philosophy that produces ‘results’ as naively defined, call it ‘science’, and then go on complaining that philosophy never produces any results—which those of us having paid attention in their philosophy classes on sound logical reasoning will recognize as the informal fallacy of ‘moving the goalposts’).
But actually, the best argument for philosophy and its importance, to me, is that it does produce concrete, visible results, that in fact are so entrenched within our everyday world that, like air, they’re all too easily missed. It doesn’t matter if you take politics, society, ethics; ideas like democracy, universal human rights, the social contract; or developments like universal suffrage, equality between men and women, or the recognition of animal rights—if you go back far enough (or just a little, in some cases), you’ll find them discussed in dusty tomes of philosophy long before they really enter into everyday life.
However, once they have, it becomes hard to imagine that things might be different; and the flipside of that is that, steeped in present society, it needs some training to see beyond its borders and question its tenets—training that the study of philosophy provides. The philosophical argument, all the while, moves on to topics that may seem as bewildering to the average Joe today as the idea that women might vote to the 18th century farmer, or the notion of democracy to the medieval aristocrat. (Of course, that’s an idealization; ideas never really develop in quite such an orderly, linear fashion, but it’ll do as a first approximation.)
So, those who typically question the value of philosophy do so while enjoying their social liberties, their right to vote, their social security, their equal rights, and so on, all the while wondering—but what has philosophy ever done for us?
(No, I’m not saying that philosophy was solely responsible for all the social change for the better in the world. I’m also not saying that there isn’t plenty of stupid philosophy around, the above ‘simulation argument’ being a case in point. I’m admittedly exaggerating somewhat, but only to counterbalance the all-to-prevalent ‘philosophy is just navelgazing eggheads’-point of view.)
One thing that I think everybody should take away from philosophy is a willingness and ability to question one’s assumptions, and recognize especially the unspoken, implicit ones. In some cases, this doesn’t get us very far, perhaps: if we ask, how valid is it really that we believe there is an external world, and we find out that it turns out we don’t really have good justification for that, then well, this doesn’t really help us terribly much—except, perhaps, in building value systems that make life worthwhile even in this case, as Mijin alluded to. But once we started questioning how valid it really is that women are inferior to men, we started building a better, more just society (a process which, of course, remains unfinished to this day)—so, while sometimes you don’t really get much of objective value, occasionally, you hit upon a game-changer.
And without many experiments perhaps coming up with nothing more than another tome nobody reads (but that at least got somebody tenure, maybe), we’d never hit the occasional paydirt.