Promos are how we get into the heart of a character. We often know his motives, and reason for behavior, with logic.
Take CM Punk, for example. People praise his mic skills, and for good reason. But when he won his Slammy two years ago, and then attacked Randy Orton at the Rumble, how would we have understood his behavior without the actual promo? As a whole, wrestling fans have short memories, and many may not have remembered when Orton, a heel at the time, punted Punk, a face at the time, and cost him his first championship.
So what do promos add? They add the story to the storylines. Often times, what's done in the ring tells a great story, but the promos tell the 'why' as to what they're fighting for. Sometimes, it's as simple as one man has a championship and the other has it, but other times, it's a deeply personal issue. Imagine how different some of the great feuds of recent, let alone all-time, would have been different without promos to sell them.
Wrestling is Sports Entertainment. When done right, such as with the CM Punk's, the John Cena's, and Chris Jericho's of today, it's a big part of the entertainment value of the sport.
When you watch the NHL, NBA, MLB, or the NFL, teams hold (some)pre and post-game interviews and sessions with the press. What are they doing? Adding depth and character and connecting with the fans of their city. Often times, especially with teams that are rivals, they say things about each other to further that rivalry. Often times, they're used as bulletin-board fodder for their opponents. How is that different from wrestling, other then being real? While we have our hometown favorites and such, promos help us establish our favorites and those we revile. Actions and words, actions and words, they're what shape us as human being, correct? It's simplistic, but its the basic logic of who we are.
It's the same thing with wrestling. It's both the actions within the ring and the words they use that shape them as characters. And that's where the enjoyment of promos come in. They help provide motives into actions and depth into characters.
While there's no cure for an ailing product then wrestling matches itself, having a reason as to why they're fighting explained well through promos provides the complete product of entertainment within context of wrestling, and when done well, makes it so enjoyable.