Holding hands, for me, is not meant simply a superficial sign of affection to the masses. In fact, rarely do I hold hands with people in public. I don't deny there are people who do this, by any means, I'm saying that is not what I do it for, nor is it the reason I like it.
When you hold someone's hand, it's a comfort contact. The way a hot bowl of soup is a comfort food on a cold day, holding hands is a way of showing affection in a way that isn't abrasive or overly sexual, but comforting. It's intimate, nonetheless. It's not meant to display your affection to the world neccesarily, or prove anything to your partner, but is a gentle way of sharing yourself with someone else.
Sometimes, when words fail to achieve the desired affect, just holding hands tells the message you're trying to portray. "I'm sorry," "I understand," "I'm glad to see you again," "I'll miss you," "I love you," "It's okay"...It can say any of these things and many more. It just lets someone else know that they are important, and what they may be going through is also important, even if you can't find the right words to tell them.
If anyone still doesn't understand how much meaning falls upon holding hands, I recommend seeing episode 5 of Kareshi Kanojo no Jijyo. If you don't understand by then, you never will