The story really begins when a mecha lion plummets from the heavens in front of a young couple in a mini-bus. While this thing is 20+ meters at the shoulder all they notice is that it is proferring a baby in its mouth. Naturally they adopt the child and the lion zooms off into the the sky. Nothing special there, happens all the time.
Anyway, cut forward 8 years. And allow me some interpretation because it's not entirely clear what's really happening. However the important part is that earth is under attack from hostile alien monsters who have a huge undersea base. They have the ability to infect humans with a alien alloy turning them into machine creatures. These creatures can then merge with other machinery and morph into monstrous mecha with awesome destructive powers. These mecha also tend to be shaped, and act on, the obsessions of the human who was chosen as the basis. Likewise each of the four `monster lords' seem to have different tastes in who they pick and how they appear. One likes flying things, one water craft and so forth. All in all a limitless number of super-powered mecha with exotic powers and appearance ready to stomp Tokyo flat.
And our only defense is the obligatory secret super-science organization, who also have an undersea base. They've got a wider variety of characters and even wider variety of mecha and gadgets. However at the heart of it all is Kai, a guy who has been re-born as a cyborg and is at the heart of Gaogaigar, their most powerful weapon. As the story progresses two AI mecha join the team. In addition the child's powers awaken and while all their secrets (or plot devices) are unknown it allows him to detect the monsters and is the only way to retrieve the human from the alloy. He, and especially his school friends, also have a tendency to be endangered by the latest mecha beast. I also forgot to mention that his `lion' also seems to be at the heart of the human super-science and gaogaigar itself.