Why water cools from outside to inside?
This has everything to do with heat conduction, Setting aside freezing or melting for the time being, the two things that are happening in combination are (1) heat conduction and  (2) the capacity of a parcel of solid to store thermal energy by its temperature changing.  Heat conduction is driven by a spatial gradient in temperature, with heat flowing from a hotter region to a colder region.
So if the outside surface temperature of an object is suddenly increased, the interior temperatures will not change at all initially, and the temperature gradient will be zero everywhere, except at the surface, at which it will be very high. This will cause heat to begin flowing into the very outer layer of the object.  The heat that flows into this outer layer will cause its temperature to begin rising (to store at least part of the heat).  This temperature rise will then create a temperature gradient for heat to begin flowing into the next layer inward, where its temperature will begin rising.  This kind of sequence continues until, later, the temperature at the very center of the object begins to change.  So, as a result of conductive heat transfer, the temperature rise propagates like a wave traveling inward into the object.
This can all be described mathematically and modeled by the so-called transient heat conduction equation.