It is pretty obvious that water can hold up water. You can test your guess with some water and two drops of different food colorings. Now, back to salt water. If you weigh a cup of water, and then weigh a cup of salty water, it turns out that a cup of salty water is heavier. The more salt you dissolve, the heavier is the cup of salty water.
Salty water can hold up salty water, that is it can hold up heavier stuff than plain water can.
An egg weighs more than an imaginary egg made of plain water, but it weighs less than an imaginary egg made of salty water. So salty water can hold it up, and plain water cannot. That is how it works. Because Dead Sea is much more salty than most world's oceans, and we are made mostly of water you can float in the Dead Sea without even trying.