The magnesium gives two electrons, but the chlorine receives only one electron. Hence 2 chlorine atoms are required to take magnesium's electrons. Now, magnesium has lost 2 electrons, and gets "2+" positive charge, as it has two protons more than electrons. And each of the two chlorine atoms gets a single "-1" charge, as they have one electron more than the protons.

Now, as positive and negative attract vigorously, the two elements combine (one magnesium atom and two chlorine atoms) to form the compound magnesium chloride, that is MgCl2.

This is an ionic bond formed by the donation and acceptance of electrons, and each of the atoms have charges.

Now, when this is dissolved in water, the water molecules attract these ions and force them to separate. This happens in almost every ionic bond when dissolved in water. When they are separated, they have charges and are in the form of ions, and hence conduct electricity, by the movement of these ions in the solution.