Why is the kitchen sink drain overflowing?

If your sink is below the sewer line elevation you need a holding tank which then is evacuated by a pump that emulates the needed lower elevation of the waste line.

Do you hear the pump motor running? If not the pump may have failed and/or it's not getting power. If you do hear the pump, but the water is still sitting in the basin the plumbing maybe clogged and/or the one-way valve to prevent back flow where dirty water can enter back into your sinks basin which you don't want.

You do need to be very careful as sewer water can carry some nasty bugs! Make sure you wash down the area with bleach as well as anything that could have been splashed and your sponges and cloths!