Hullin, Chapter One, Mishnah One
Our mishnah teaches some general rules with regard to who may slaughter and when.
1) All may slaughter, and their slaughtering is valid, except a deaf-mute, an imbecile or a minor, lest they mess up [the animal] through their slaughtering.
a) And if any of these slaughtered while others were standing over them, their slaughtering is valid.
2) That which is slaughtered by a non-Jew is a nevelah and defiles by carrying.
3) If one slaughtered at night, and also a blind man that slaughtered, the slaughtering is valid.
4) One who slaughtered on Shabbat or Yom Kippur, even though he is liable for his own life, the slaughtering is valid.
Section one: Anyone may slaughter an animal. The only exceptions are the three categories of people who are not considered to have daat, intelligence. They are not allowed to slaughter and if they do slaughter, the animal is not kosher. The problem is that one cannot tell if an animal was validly slaughtered just by looking at it afterwards. Thus there is no wa to check the results if one wasnt there watching when the animal was slaughtered. Furthermore, these three people cannot be trusted to report on what they have done, because they are not considered to have intelligence. Therefore, their slaughtering is invalid. However, if an adult is watching them while they slaughter and sees that they do it correctly, the slaughtering is valid.
Section two: A non-Jew cannot ritually slaughter for a Jew. If he does, the animal is considered a nevelah (carrion) and it imparts defilement by being carried (as do all nevelot, see Leviticus 11:40). The mishnah probably notes that this meat defiles in order to teach that this animal is deoraita (by the Torah) considered a nevelah, and that this rule is not just a stringency initiated by the rabbis.
Section three: Obviously, it would be better not to slaughter at night, and it is problematic for a blind person to slaughter. However, an animal slaughtered at night or by a blind person is still valid, as long as it was slaughtered properly.
Section four: Slaughtering is a prohibited labor on Shabbat and Yom Kippur and one who does so is liable for the death penalty (Shabbat) or karet (Yom Kippur). Nevertheless, the animal is still kosher. Just because the person sinned, doesnt mean he didnt slaughter in the correct way.