Specific Heat Derivation
If a body absorbs a quantity of heat $q$ its temperature will normally rise by a value $\Delta T$. The average heat capacity  over this temperature range is defined as $C_{av}\equiv q/\Delta T$. The instantaneous heat capacity at temperature $T$ is $C\equiv dq/dT$. This definition is not exact enough, however, until the path of heating is specified. From the first law $q$ could be replaced by $dU+dw$ so it is necessary to fix another parameter which is $V$ or $p$ giving rise to the definitions you quote.
Note (a) that at a phase change $C$ becomes infinity because heat is absorbed with no change in temperature. (b) As the heat capacity is the gradient of internal energy with temperature it describes how the internal energy increases. For example when $C$ becomes constant at high temperatures the molecules are absorbing heat a constant and maximum 'rate' with increase in $T$. (c) Using statistical mechanics the heat capacity can be calculated from a knowledge of a molecule's energy levels.