I too am fascinated with the number 13. My birthday is on February 13, but I think my attraction to the number extends beyond that, and even beyond much of the cultural myth around the number. It is a very powerful and magical number, it seems. I think the fact that there are thirteen moons in a year, which others have pointed out, is part of the key to the power this number has held in the human psyche.
It is a prime number, a number which cannot be divided by any other whole numbers besides one and itself. It juxtaposes two other numbers which hold great power over the human mind - one and three. The two digits add up to four, which also has much power over human thought. Three is a sacred and powerful number universally, from the Holy Trinity of Christianity to the Maiden/Mother/Crone forms of the Goddess. One is a number which points to a higher power, something essential. One can only be divided by itself, and all other numbers can be divided by it. Four is a number which represents wholeness and stability fairly universally. We have the four directions, the four seasons, the four elements (if one does not count "spirit" as one of the elements), the four corners of a square. In these three numbers, we have three powerful and symbolic shapes, going from number of sides: circle, triangle, square.
I also think that something else others have touched upon is key to understanding the power of thirteen. Another powerful number in the human mind is the number 12, which recurs cross-culturally as an important number. The twelve months of the year, twelve signs of the zodiac, twelve tribes of Israel. The number 13, I think, implies the fullness of the number 12 as well as one added element. I think the sense of magic in 13 at least in part comes from the perception of having twelve concrete objects, and then one more thing that is essential or underlying, something mystical or profound which unites those twelve. We cannot see this thirteenth element, but it is there.
I think this is also why the number 7 and the number 5 also have such a magical sway. We have twelve, or six, or four elements, divisible and balanced, and then add one more, something intangible that does not throw the other elements off balance but adds something powerful. This added element makes the number indivisible, and thus make it seem to stand on its own, undivided. A good example of this is the idea that there are five "magickal" elements: fire, earth, water, air, and spirit. The first four are tangible to some degree. The fifth is central, essential, but intangible.
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