Why is there no apostrophe when saying something belongs to it? Its foot fell off, rather than It's foot fell off.?

NONE of the possessive pronouns has an apostrophe in it. You wouldn't write any of the following, would you?

He'r foot.

Hi's foot.

Thei'r feet.

Ou'r feet.

M'y foot.

So why would you put an apostrophe in the possessive pronoun "its" when you don't put it in any of those other ones?

The apostrophe in possessive forms of nouns (John's, the president's, the computer's) comes from a historical source -- it's a contraction of phrases like "John his" -- which is not applicable to the possessive pronouns.

"Its" is a possessive pronoun. Pronouns do not have apostrophes. We don't say "our's" or "your's" either.

The way to remember is that "it's" with apostrophe is ALWAYS a contraction for "it is."

For the same reason their isn't an apostrophe in the word "his," or "hers," or "my," or "theirs." It's just the way it is.

"Mary" is a name; the possessive is "her." If you wanted to, you could say "she" instead of Mary. "It" is not a name; it's a pronoun, as "she" is. "Its" is like "her": both possessive without apostrophes.