Every complete sentence contains two parts: a subject and a predicate. The subject is what (or whom) the sentence is about, while the predicate tells something about the subject. To determine the subject of a sentence, first isolate the verb and then make a question by placing "who?" or "what?" before it -- the answer is the subject.
Your sentence "YouŽll never get it now!" said Swiper as he flew away. could be rewritten thus: Swiper said "YouŽll never get it now!"as he flew away.
This way it is is easy to see that said is the verb and Swiper is the subject. The rest of the sentence including the clause "YouŽll never get it now!" is part of the predicate.
In your sentence "I was outside looking at the stars when a flying saucer fell from the sky" was is the verb so I is the subject and the rest is the predicate.