Leaves keep breathing after they are picked. In an open fridge they lose water fast, and that is what wilting is.
Before they go in the fridge
Do not wash greens until you are about to cook them. Shake off any soil, pull off yellow leaves and untie the bunch so air can move between the stems.
The damp cloth method
Wrap the bunch loosely in a clean, damp cotton cloth and put it in a box with a lid. The cloth keeps the air around the leaves humid without leaving them wet.
Wet leaves rot. Damp air keeps them crisp. That is the whole trick.
What keeps and what does not
Chard, spinach and methi keep three to five days this way. Lettuce keeps four days in the box it arrives in. Coriander does best standing in a glass of water, like flowers, with a loose bag over the top.
If they have already wilted
Stand the stems in cold water for half an hour. Most greens come back. If they do not, cook them: wilted leaves are fine in dal or a paratha.