I've seen this argument innumerable times. In the end, this is like asking why you are happy when your team wins. After all, you win nothing and your life stays the same.
Reasonable fans of sport clubs care about how the money is spent because spending money has consequences for the club they support. If your team spends too much on a shit signing, you worry, not because it's your money, but because it hampers the club's capacity to stay strong and competitive, and you care about that. And even if it's not a shit signing, if it compromises the sustainibility of the club, it's reasonable to be worried if you're a supporter.
I guess this is not news to you, but football clubs don't have endless resources. Valencia is a recent example of how bad management can compromise the viability of a club. Or look no further. Barça under Gaspart was in serious trouble, in great part due to crazy spending. Meh, who cares, it's not my money!
That said, I'm ambiguous regarding the signing of Arda Turan. Short term, great. Long term (thinking about our young prospects) I'm not so sure. I'm not the least worried about the money spent, but obviously not because it's not my money, but because I think Barça can currently afford it.