The thing about movie is it doesn't need a complex plot but it should deal with a lot of complex emotions.
The story as a whole could be about the first time Bowser kidnaps Peach (forget Partners in Time, he just does this as an adult) and the not-yet-heroes Mario Bros. are thrown in a situation where they have to save her.
Simple plot but think about this.
While this is going you see the relationship between Mario and Luigi. On the surface they seem to get along, but during the movie you find they have underlying issues with each other.
Mario secretly looks down on his younger brother, feeling like he's holding up the plumbing business all by himself. Luigi is jealous of Mario but also sees him as arrogant. They eventually fight, but throughout the trials of the movie they learn how much they love each other, and that they need each other to pull through. Then you get some really heartwarming scenes.
But that's not all!
Bowser has captured Peach, on the surface he looks like a monster, but throughout the movie you learn he is insecure and has no idea how to help his people and children who live in the horrible Dark Lands. Peach shows some sympathy for him which eventually makes Bowser fall in love with her. Peach does not love him back and Bowser blames Mario who is currently fighting his forces and is making him feel weak and pathetic.
See there's a lot going on here, even with the simple plot of rescue the princess.