After a tree has been felled, the root network will continue to feed the stump. A living stump will not rot and may grow new shoots. Epsom salt (or magnesium sulfate) is hygroscopic, which means the crystals absorb water. In sufficient quantity, Epsom salt pulls moisture from the wood, which then kills the tree.
There are many substances that could be applied to a tree stump to kill it, but Epsom salt has advantages over other stump removal chemicals. Rock salt or caustic lye would dry out the wood, but high sodium levels could make it difficult to replant after the stump has been removed. Commercial stump removal preparations contain potassium nitrate, which aids decomposition but does little to kill a living stump. Epsom salt kills the stump while improving the surrounding soil by adding magnesium and sulfur; plants require both for chlorophyll production.