Where the earth's crust gets thin that part of the surface sinks downward. Look at the picture below to see how the rift forms, sinking downward where the crust is stretched thin. The Dead Sea is still sinking lower, even today. Scientists figure that the Dead Sea lowers by as much as 13 inches per year. Anyway, back to the formation of the Dead Sea.
All roads lead to the Sea when it comes to the rivers in the area.
The Dead Sea is continually fed water from the rivers and streams coming down off the mountains that surround it. The only way water gets out of the Sea is through evaporation. This part of the world get plenty hot. When the water evaporates, it leaves behind all the dissolved minerals in the Sea, just making it saltier. The fact that the water doesn't escape the Sea just traps the salts within its shores.