To know about why ice stick to your skin we have to first know about the formation of ice. On the outer level we say that when we cool the water to certain temperature it turns to ice but on molecular basis it is something different.
We know that the water (H2O) is made up of two molecules of hydrogen and one molecule of oxygen. But oxygen has the two extra pairs of electrons which are known as lone pairs. And they make the molecules of water a tetrahedron.
When one of the lone pairs of oxygen goes out and attaches with the hydrogen atom of the other neighboring molecule then the interaction of molecules start which is known as the hydrogen bonding. Every molecule in the water is attached with the nearby molecule through hydrogen bonding. So when the water is in liquid state every molecule change its interaction and move on making new connection with new molecule which results in water flow. But when temperature is very low in the case of ice it cannot break old connection and form new one. Now the molecules are not in the condition to change their interactions so it remains on one place forming a three dimensional hydrogen bonding.
On the surface of ice some oxygen atoms and hydrogen atoms are left facing outwards which causes them not to connect with the other molecules to form hydrogen bond. So they are looking for a partner to form the bond and when a surface that is able to form this bond comes in contact with the ice surface it will stick to that, like our skin. Our skin has high moisture content and very appropriate to form the bond with these atoms that is why it sticks to your skin.