I shall to try to get across the basics here. A metorite crashes to earth so far pretty simple... A princess is on her way to Japan... OK... Then her plane is destroyed but she shows up claiming to be a Martian. Ok what did I miss?!
The Martian Princess predicts doom and destruction and the bad guys are out to finish the job they bungled when they blew up her plane. (I told you it makes no sense) Sigh. Then the meteorite cracks open and out comes Ghidrah! Yay!
Ghidrah proceeds to go to Tokyo (What did I tell you? A monster magnet!) and starts to lay waste. What shall we do? Rodan and Godzilla are fighting each other (and enjoying it by the looks of things) and Ghidrah is just tearing up everything!
Oh look, here come the twin faries of Infant Island. they can get Mothra to help, right? Well Mothra is still a larva from the last movie and she won't be much help. But wait, maybe she can get Godzilla and Rodan to quit fighting each other and team up with her! Yeah, thats the ticket!