Alexa V. answered 05/23/19
10+ Years Translating Japanese-English with JLPT N2 Certification
The implied subject here, "I," is being omitted. In Japanese, the subject can be removed when the meaning of the sentence is clear from context. To answer your ending question, Sentence 1) means "I like cats." and Sentence 2) means "Cats like me."
は and が are difficult particles to differentiate in the beginning of your Japanese studies. For simplicity's sake, think of は as the "subject marker" for now: the subject of the sentence is the noun preceded by は. The learner you asked is explaining this in an abbreviated way by putting 私は in parentheses before the sentence in question (猫が好き).
When using these sentences in real life, the subject might be assumed based on context. If you're taking about your likes and dislikes, 「___が好き。」would mean "I like ___." Meanwhile, if you're talking about your sister, the assumed subject would result in "My sister likes ___." even if you just said 「___が好き。」.
Let me know if you would like a more detailed answer. Happy to help!