To understand this phrase, think first about its literal meaning. If you were kicked by a mule, you would be hurt. So, in theory, you would not want to be kicked for a second time because you would remember the pain and general unpleasantness of the experience.

To give an example of this phrase in action, let's imagine that you do not study for an important test and you fail it. If you are given a second chance to take the test, consider whether or not you would study for it. Would you let yourself fail again by not studying (and, therefore, be kicked by the mule for the second time), or would you study and make sure that you pass?

Looking deeper, then, this phrase encourages you to learn from your mistakes and, more importantly, to not make the same mistake a second time. 

The meaning of this phrase is sort of similar to the idea of "fool me once, shame on you.  Fool me twice, shame on me."  It means that the first time that something happens to you, you should learn from it.  If it happens a second time, it's your own fault.

We see this phrase used a lot in politics.  I have heard Republicans say that they will not vote for President Obama's most recent jobs bill because (they say) we have already tried those things and they do not work.  If we do it again, it is our own fault.  In connection with this, they use your saying about the second kick of the mule.

In your personal life, it could apply to many things.  Like maybe it would apply to not studying before a test or to going out with a guy hoping to change him.  It's basically applicable to any situation in which a person might be in danger of making a mistake for a second time.