How Does The Presence of Livestock on Land Increase Net Methane Production?
Ruminant digestion produces a lot of methane.
Cows are ruminants, and ruminants produce a LOT of methane. Sheep and goats are also ruminants but their are far fewer of them. Ruminants use methanogenic bacteria to breakdown their food, these bacteria, as there name implies release methane as a byproduct.  Ruminants produce a lot more methane than other herbivores becasue of this.
Decaying grass normally produces CO2 as do many native herbivores. Cows like most herbivores only convert a very small amount of the grass they eat into cow. Herbivore digestion is not terribly efficient, not surprising the grass does not what to be easy to digest.
Farming also puts a lot more herbivore mass on a piece of land than it would normally have, further increasing the effect, that is the purpose of farming to get more out of the land than simple hunter/gathering could. This is an issue becasue methane is a FAR more effective greenhouse gas than CO2.
Feeding cattle often involves replacing existing ecosystems which high yield feed crops and this can release a lot of CO2 and/or Methane depending on the original ecosystem just from the change, but it also means that feed is now producing methane instead of CO2 it would be if eat by non-ruminants.
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