I am working on a project in my chemistry class, and even though we've already finished the lab, I'm still confused as to why and how $\ce{MgCl2}$ melts ice/ lowers the freezing temperature of water. Can anybody tell me why this happens?
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The $\ce{Mg^{2+}_{(aq)}}$ and $\ce{2Cl^{-}_{(aq)}}$ ions disrupt hydrogen bonding, preventing ice from forming a lattice structure. Therefor, the water must be colder so those weak hydrogen bonds will be acting on water molecules with less kinetic energy (i.e., colder), finally allowing them to create a crystal around the dissolved "impurities" $\ce{Mg^{2+}_{(aq)}}$ and $\ce{2Cl^{-}_{(aq)}}$.
This is a basic colligative property. Its more effective for electrolytes because they dissociate, creating more lattice disruptions. As a general resource, see: Ebbing, Darrell D., General Chemistry, 3rd ed, Houghton Mifflin, 1990.