Words express our feeling about surrounding but actions shows them. Words are only reflection of action. Words can lie and so aren't as trustworthy. We make promises without thinking and latter break them.
Everyone can say whatever they want. We say we will do this and do that and be this and be that. But unless we actually go out and try to achieve it, what's the use ? Usually people evaluate a person, understand them, get to know them by their actions and not their words.
An example can be made of the politicians. When the elections are near and they are giving speeches on how they will help improve our nation. Their promises are just words. And they remain words. If they put their promises in action, then maybe the people will not always be trying to lynch them. Their actions speak for them.
Hence the proverb means that what someone does is more important than what he says.