why not "Ich bin eine Frau die Katzen mag" ?
Sollte auch richtig sein. Das ist eine Frage der Intensität.
But you need a comma after "Frau".
Referring to a substantive in the main sentence through "der, die, das" is done with a subclause (Nebensatz). The comma seperates the two.
mögen is to like -- this sentence uses "love", which is stronger than just "like".
Is this 'liebt' rather than 'liebe' because in the sentence it's technically 'the woman' who loves cats and not 'I' - even though I'm the woman?
Similarly, even a man would say Ich bin eine Person, die Katzen liebt with the feminine relative pronoun die, because it applies to the feminine noun Person rather than to the ich that refers to a male person.
die katzen liebt order die liebt katzen und warrum?
It's die Katzen liebt with a capital letter on Katzen.
die Katzen liebt is a relative clause, and those are a kind of subordinate clause -- and thus the conjugated verb (here: liebt) comes at the end in German.