A huge source of arguments, intelligence gathering, stress, gloating, and smack talking. In my old corporation they didn't matter at all to us. The CEO and directors didn't monitor them to make sure that everyone maintained a certain ISK efficiency or anything like that. People still looked at them and asked questions about kills and losses, but nobody was yelling at you for losing ships. Maybe if you lost a billion worth of loot in an Iteron V in high sec, you might get a few extra jabs for being stupid, but we didn't even take those too seriously because the person that lost the ship already has to deal with the loss as well. Apparently that isn't the norm though, because I've been in a few alliances in my eight months of playing Eve and most of them are pretty serious about their killboards and try to make sure people don't die too much.
This is a strange catch-22 in my opinion. They want you get out there and fight and blow up other ships, but to do so you have to risk your ship, which means that you are going to lose ships too. Since I moved back out to null sec, I have killed around 2-3 billion in ships, and I've only lost one Talwar for about 13 million ISK. I think that's pretty damn good if you ask my opinion, but for some reason people always focus on the negative.
A few days ago a corp mate in his Thorax and me in my Talwar took on a Stabber Fleet Issue (SFI) and we both lost our ships in an EPIC fight. We had the SFI down to about 25% hull before we died, and it was glorious. I was not upset in the slightest from losing my ship, neither was the Thorax. The SFI had 90% overheat damage by the end and thanked us for an awesome fight. He scooped up both of our wrecks, and contracted them back to us for free because he had so much fun with the fight. Nobody was upset at all in the end except the CEO that we lost. Why was he upset? Not so much that we didn't win, but because with the Thorax and Talwar deaths, we had now dropped to 89.9% ISK efficiency on our killboard.
He didn't yell or get angry or anything, but you could hear the heavy sighs and disappointment. I felt like saying "Sorry Dad, I'll try harder next time", but I didn't. I just let it go. When I was a kid I remember always feeling like I was being told what I was doing wrong, and never told when I did something good. Reason is, if you're doing your job right and not making mistakes, then people in authority positions (Parents, Bosses, Teachers), will typically leave you be and you never hear from them. The moment you do something wrong though they become "Johnny on the Spot" and are there to remind you that you're doing it wrong. This is why I make an active effort to acknowledge the successes my 10 year old son has as well as his mistakes.
So why do killboards matter? Whether you use Battle Clinic or Eve Kill or some other board, why do they matter in the end? I know that some groups won't allow you to join if you don't have a certain number of kills a month or aren't operating at a certain efficiency level, and I'm sure that's because they don't want someone joining their group that can't hold their own. Other than that though, who cares if you die? Ships are meant to be blown up in Eve. You're going to have good days and bad days, good months and bad months, Eve is a game of give and take. An ex corp mate used to fly sniper fit Cormorants and he'd lose about three or four a day, but he also killed about 40-50 other ships a day. That's a solid trade off in my opinion.
I try not to take Eve too seriously. I enjoy it, it's a lot of fun, but in the end it's still a game. I could shut it off tomorrow, walk away, and New Eden wouldn't even notice I was gone.