Grounding sheets are intended to replace the direct contact from the earth to your skin that you would get if you walked barefoot. For most of human history, walking directly on the ground was common, but that has changed in modern life.
The earth’s surface has a slightly negative electric charge. Because of reduced exposure to nature and increased exposure to electronics and EMFs, the body often has a slightly positive charge. When it comes into direct contact with human skin there is a balancing of the charge. Electrons reduce the positive charge in our bodies. Negative ions have been found to play a role in maintaining health in this way.
 A grounding sheet connects you to the electric current of the earth and as a result, lets you bring the experience into your home.

The exact mechanism involves some theory, but there are studies showing numerous positive effects of grounding, so the science is there. Studies have provided evidence that grounding normalizes cortisol levels, promotes healing through reduced inflammation (fewer free radicals), helps with pain relief, promotes better sleep as well as numerous other wellness benefits. Note that we're summarizing the literature, not making product claims. Read the summary of the literature linked above to draw your own conclusions.

Assuming you've used a continuity tester to verify the functionality of the sheet and a socket tester to ensure your outlet is properly grounded, it is a matter of observation of your experience. Writing a journal of your experience helps you keep track of things that are changing. If you experience pain, for example, record your pain level each day as well your subjective experience and look for any changes.

Many people report that it can take up to 30 days of consistent usage to notice the results. The amount of time you typically spend sleeping should be plenty of grounding time per day.

Improper care can cause a reduction in the lifespan of the sheets so you do need to care for the material properly. Most problems come from the use of bleach, fabric softeners or harsh detergents that can leave a heavy residue on the sheets, reducing their conductivity.
It's fine to put them in a normal washing machine but it's best to use a laundry detergent that is gentle, a free and clear detergent is a good choice. You can also use white vinegar as a gentle detergent. It's also fine to line dry the sheets or to use a dryer on a low setting. As with all fabrics, gentle cleaning will extend the life of the product and you just want to avoid anything that will leave a film on the conductive fibers.

These sheets have roughly the same lifespan as normal sheets, provided you follow the instructions.

Researchers have recorded positive effects of using grounding to enhance the depth and length of sleep, reduce pain, and reduce stress. The first study was published in 2004 and found that grounding improved sleep and reduced levels of cortisol in the body, which is a stress hormone.

The sheet is 5% silver conductive thread (the composition of which is 18-20% actual silver), the remainder is organic cotton and the elastic that holds it fast on the bed

Yes it does, by the Global Organic Textile Standard.

When used as directed, they are as safe as any other sheet.

The research indicates that there are effects of grounding, but we are not aware of any negative side effects of using grounding products. It is intended to mimic the effects of being in direct contact with the earth, which is not inherently harmful in any way.