Assume an earthlike planet with less mass than Earth. The planet is super-habitable. It has shallow oceans, was geologically active but is now geologically inactive, the main continents are east-west and there are no north-south continental land masses like Africa and the Americas.
The planet stopped being geologically active approximately two million years ago. Vulcanism ceased, i.e., no volcanoes. Continental drift stopped. Mountain building was halted. A planet with shallow ocean will have warm seas which pump enormous volumes of water vapour into the atmosphere leading to
high rainfall globally. The mountain ranges have been ground down to effectively nothing soil erosion and water activity over the last two million years. Much of the shallow oceans has filled with silt and run-off soil.
Water levels have risen to cover large areas of the mostly flattened continental land masses.
Marshes and swamps are now the dominant land form. There will be areas where the free oceans still exist. It seems worthwhile to leave some vestiges of the planet's earlier environment. This adds to its plausibility. And it adds to the author's plausibility: I, the author, know one-environment planet aren't too likely. Note: there are some exceptions, but we don't need to worry about them.
Because the planet is, was or has been super-habitable, there will be vast reserves of material to drive the food chain (this could be more difficult than this glibly suggestion; but it can simply be assumed it is so) though this may be mainly detritus feeding, which is much like swamp or marshland ecology anyway.
This means there should be a rich flora and fauna on the planet which as it changes into Planet Marshlands adaptive changes and evolutionary pressures will adapt them to this environment.
The above model is based on the super-habitable planet and is well worth your while to do some further research into to help build your planet. The Wikipedia entry linked above is a good starting point.
There is a certain amount of pure hand-waving in this model for marshlands-only (well, almost) planet There is another putative model for a swamp environment planet, but that involves a super-Earth type of planet with high gravity and a rigid crust which suppresses mountain building, vulcanism, and continental drift.
In this case, the high gravity would be the killer. No natural Earth humans could move around such a planet without lots of advanced technology. While this logically makes sense, it's the death knock for plausible fiction.