Want to eat meat? Certain animals are okay, while others are right out. Kosher animals must have both of the following signs: they must chew their cud and have hooves that are completely split. (This is spelled out in the next verse, 11:3.) So a cow is okay but a horse simply isn’t kosher. A deer is all right, but a pig won’t do. How about a gazelle? A giraffe? An armadillo? An ibex? A capybara? (That’s yes, yes, no, yes and no, respectively.) How do we know? You have to check.
From the fact that the Torah specifies “these you may eat,” we can easily infer that “others you may not eat.” This is called a “lav haba m’klal asei,” a prohibition that is inferred from a positive commandment. (We’ll see more about non-kosher animals in the next mitzvah.) The reason for the prohibition is the same as the reason for prohibiting animals torn in the field and certain fats (Mitzvah #73 and Mitzvah #147, respectively). God knows that certain things are spiritually bad for us, so He warns us not to eat them. We are obligated to inspect animals for the signs of kosher animals in order to ensure that we only eat that which we should.
There are other signs that are unique to kosher animals, as well as differences between kosher wild animals and kosher domesticated animals. (This is important because there are differences between wild and domesticated animals as to whether or not the blood must be covered after slaughtering and which fats may or may not be permissible.)
This mitzvah applies to both men and women in all times and places. It is discussed in the Talmud in tractate Chulin (59a-b) and codified in the Shulchan Aruch in Yoreh Deah 81. It is #149 of the 248 positive mitzvos in the Rambam’s Sefer HaMitzvos. The Ramban does not consider this a separate mitzvah, considering it merely a necessary prerequisite of not eating non-kosher animals. Similarly, it is not listed independently in the Sefer HaMitzvos HaKatzar of the Chofetz Chaim.