The bosses at Toho films weren't very pleased by Godzilla vs Hedorah and with the next film they decided to return the character to his "roots". By which they meant "fighting alien invasions". I'm pretty sure that Big G's actual roots were "Godzilla is a force of devastation that will annihilate all Japan", but I guess the whole alien-fighting thing has been his main schtick since he turned hero in his fifth cinematic outing.
We begin the film with a comic book artist who is trying to pitch his idea for a 'homework monster' to a publisher. It doesn't work so well, so a woman who is probably either his sister or his girlfriend - if the film explains their relationship I missed it - sends him to Godzilla Tower to see the people there. They like his idea, which proves pretty conclusively to me that they're the aliens in the flick.
Soon after, our not-so-intrepid hero witnesses a young woman being chased from Godzilla Tower. She drops a tape which he recovers. Later, she confronts him and they listen to the tape. It doesn't make any sense to them, being just strange noises, but on distant Monster Island, Godzilla hears it and sends armadillo-like monster Anguirus to check it out. And I do mean 'sends'. They have a conversation in Monsterese.
The Godzilla speech bubbles actually do appear on-screen in the film
Anguirus is useless as a scout though and comes back after the Japan Defence Force shoots him with some missiles. So Big G has to look into it in person.
Meanwhile, our human protagonists discover that the people running Godzilla Tower are cockroaches from outer space (yes, really), who plan to conquer the planet with the use of two space monsters. These are frequent adversary King Ghidorah, and newcomer Gigan, after whom the film is named. Gigan has a buzzsaw in his belly. It looks as silly as it sounds.
The rampage the two space monsters go on is a pretty good one, though, and by far the most entertaining set piece of the film as they blow up and smash office blocks, refineries, and any human forces that try to stop them.
Finally Godzilla and Anguirus turn up and the kaiju-on-kaiju action starts. By which I mean that Big G and his useless buddy get their butts stomped for a good 15 minutes. Godzilla even bleeds for the first time in the series. Things look bad all 'round, especially with the cockroach people firing lasers at Godzilla from Godzilla Tower.
However, when the humans manage to sneak into be building and take out the lasers, that's Big G's signal to show FIGHTING SPIRIT. He Hulks up --
<- no, not this Hulk ...
... this Hulk ->
-- and stomps the bad guy monsters in about three minutes flat. The end.
If that sounds a little underwhelming, well ... it was. Overall, I'd rather watch Godzilla vs Hedorah. At least that was nutty enough to be fun.