There is no temperature change at the freezing point because there is an equilibrium between the solid and the liquid.
For example, at the freezing point
#"H"_2"O(l)" ⇌ "H"_2"O(s)"#
in the diagram above, water molecules are attaching to the surface, and this releases energy.
At the same time, molecules from the ice are moving into the liquid water, and this absorbs energy.
However, the molecules are moving in each direction at the same rate.
Since energy is being absorbed and released at the same rate, there is no net change in heat content, and the temperature stays constant.