The argument from personal experience goes something like this:
"I know there's a lot of evidence suggesting otherwise, but I've had some experiences that prove to me that there's something out there that science doesn't know about. I realize my experience doesn't count as scientific evidence, as it is not repeatable, so I do not expect to persuade anyone. But my experience remains, and it is persuasive to me, if no one else."
This is used to argue for the existence of God, aliens, spirits, or any sort of cosmic woo-force. Well, usually, people insist that they don't intend to convince anyone else, so perhaps "argument" is the wrong word. The said experience is usually either a strange coincidence, or something like an alien abduction or Out of Body Experience.
Strange coincidences are explained through people either missing a genuine naturalistic link between events or underestimating the probabilities of coincidences. That's a topic for a future post. I'm mostly focusing on the latter type of experience.
There are two ways that the argument can be wrong. One is that the person is lying, and had no such experience. The other is that the person has somehow tricked him or herself. In my experience, the second choice is much more often the correct one. Generally, people have no motivation to lie, but are very good at fooling themselves. In other words, I think these are real experiences. People who have had these experiences really saw and felt the things they say they did. The reason I find the argument unpersuasive is because these real perceptions do not necessarily equal real spirits or aliens.
What do these real perceptions indicate, if not spirits or aliens? Generally, they indicate psychological phenomena. Alien abductions are usually explained by sleep paralysis, which is a surprisingly common phenomenon. The precise explanations for Out of Body Experiences and Near Death Experiences are sort of open questions of psychology. In a way, people are right that these experiences indicate something science doesn't yet understand. I wouldn't have expected science to have all the answers immediately, especially since such experiences are usually unrepeatable and difficult to test. But if we did find the explanation, I don't think you'd be able to understand it without a proper backing in neuropsychology. These questions are far more likely to have complicated psychological mechanisms than to have deep metaphysical ones.
Because I usually don't suspect people of lying, and because I accept the reality of the experiences, the argument from personal experience should be equally persuasive to me as it is to the person who actually had the experience. Or rather, it should be equally unpersuasive to the two of us. If I can accept that others have been fooled by tricky psychology, you can accept that you yourself have been fooled by tricky psychology. This doesn't make you crazy or deluded at all--skeptics have these experiences too. It just means that you've experienced one of the mysteries of the mind, and mistaken it for a mystery of the universe. You simply didn't realize how easily people can fool themselves.
Further reading: Testimonial evidence, as explained by the Skeptic's Dictionary.