A few people have asked me to post an explanation of Mayhem Intergalactic's gameplay. I've made a couple of attempts, but evidently I'm not doing very well.
I suspect this is because the game is very simple! Any attempt I make to explain it ends up being complicated because the most similar well-known games (e.g. Master of Orion) are complicated.
The bottom line is that the best way to understand Mayhem Intergalactic is to play it. Which, after all, is what the demo is for. I realise that people would much rather have a nice potted explanation, and are rating the game down because they're not getting it; but when the game is so different (in a good way) from people's expectations, such an explanation is hard to give.
So, please do yourself a favour and simply play the demo. It's really quick to blast through a game, it will be easier to understand than my long-winded explanations, and I guarantee you'll enjoy the experience.