Edamame soybeans are best picked when the pods have filled out & the seeds mature, but before the pods begin to dry down. The best visual indicator I can offer is to pick them when the first leaves begin to yellow in the row, and some pods begin to change color.

The exact stage for harvest is dependent upon the variety, and your taste preferences. Some with tougher skins are best picked early, while others develop their best flavor late. I just picked one today that was very good after almost all of the leaves had dropped... but that is an exception. Most varieties would have significant pod drying at that point. Test a few pods every few days, and cut all of the plants that you intend to harvest when you are satisfied with the flavor & texture. Let a few go for seed, and to test in later stages of maturity.

Preparation is easy. Strip the pods from the stems & wash them. Cook them for 10 minutes; I usually boil them in lightly salted water, but you can steam them also. You might want to adjust the time by an extra minute or two, for large-seeded or mature beans, or for larger quantities... but test one at 10 minutes, then each minute thereafter. Do not over-cook. Remove from heat promptly when done, cool slightly, and eat immediately for best flavor.

You can open the pods if you wish, but the easiest way to eat them is to squeeze the pods, and pop the beans into your mouth.

If the beans will not be eaten immediately (or if they will be frozen) they should be shelled as soon as possible. They can absorb unpleasant flavors from the hull.