Water consists of small building blocks called molecules. These water molecules don't sit still - they wiggle and move all the time. The temperature of the water is a measure of how much these molecules move. When you cool water, you lower the temperature, and the molecules slow down. As you cool the water, they move so little that they can form strong bonds and the water freezes into ice, which is a solid. For pure water, this transition occurs around 0 degrees Celsius. Conversely, if you add heat to a block of ice, the molecules will start to twist more; eventually they move too much to stay stuck, and the ice melts and turns back into a liquid. However, these transitions do not happen instantaneously - water and ice can exist simultaneously. (That is, a whole ice cube does not turn into liquid water at once).  

When the water is not pure, the water molecules cannot connect as easily to form a solid; other particles get in the way. That is why the freezing points of salty, sugary or other water solutions - temperatures at which they become solid - are lower than 0 degrees C.  

When salt comes into contact with ice, it causes the ice to melt slightly. The small amount of water from the melted ice and the salt combine together, lowering the freezing point of the water. Salt water has a lower freezing point than fresh water by approximately -2 degrees C. This lowered freezing point makes it more difficult for water molecules to recrystallize into ice.

In this experiment, when the salt was added to the ice, the ice melted a little and then quickly froze around the stirring. This causes the ice to stick to the whisk when it is pulled from the glass.

The same concept is used when adding salt to your driveway or sidewalk in the winter. The salt lowers the freezing point of water, so it melts so you can remove it more easily.