>>2010644the other anon's explanation didn't address the issue. there's two different applications of the apostrophe at work here. the first, to indicate possession, as you've identified, and the second, to indicate a contraction, as the other anon pointed out.
the important thing to note, is that pronouns never take "apostrophe s" to become the possessive form. he -> his, she -> her, they ->their. it follows that "it" does not become "it's" for the possessive form. instead, "it's" is the contraction of "it is" and "its" is the possessive form of "it".