Whether we are talking about morality or the world around us, many people think they can persuade others of their views simply by saying that the really think they are right --- that they have a strong intuition that their view is the right one. While strong intuitions can give credence to certain claims, the trouble is that if the person you are trying to convince has strong intuitions for an opposing claim, there is really no way to settle the matter.
Personal anecdotes are another way of providing support for one's claims. For example, someone might say that he believes that dogs use reason like humans because he saw his dog do something that could not be explained using stimulus-response language. While this person may genuinely believe that he saw their dog do something that could only be explained by reason, there may be no way to confirm it, since it is in the past and may never be seen again. It is true that dog intelligence has been studied scientifically, but while this has shown that dogs learn, there is as of yet no evidence that they use reason.
Empirical evidence is the strongest kind of evidence available. This is obtained through controlled experiments in which a claim is tested. If a claim is corroborated many times over, there is good reason to believe in it. For example, there have been an uncountable number of attempts to counter the theory of evolution by natural selection. However, every time evolution is tested, the results come up in its favor (in the fossil record, for instance). Claims are convincing if they are backed up with solid data.
One may wonder whether there can be empirical evidence in support of moral claims. While it is a contentious issue, it should not be ruled out. Consider the claim that hitting someone for no reason is generally bad. If we can produce brain scans showing that areas of the brain associated with the experience of pain become active when a person is hit, we have evidence that the person feels pain, which is generally thought to be bad. This would seem to lend credence to the moral claim that hitting someone for no reason is bad. Of course, morality is also a product of societal norms. In order to know that it is wrong to hit someone, we have to accept that inflicting pain is morally wrong. Nevertheless, this example does lend empirical support to a moral argument.