How can energy be absorbed without changing the temperature? I'm a newbie in physics, so I went reading on wiki to learn about heat of fusion but then it said how $1 \:\rm kg$ of ice at $0^{\circ} \rm C$ under wide range of pressure needs $333.55 \rm kJ$ energy to fully melt it. See:
https://en.m.wikipedia.org/wiki/Enthalpy_of_fusion (first paragraph)
So if the ice absorbed energy to melt, shouldn't the kinetic energy of molecules in the substance increase, and thus increasing the temperature? Why is the temperature constant at $0^{\circ}\rm C$ while energy is absorbed?
 A: What does internal energy depend on?
The notion that the internal energy depends only on the temperature is wrong for any real gas/liquid/solid. The fact that internal energy only depends on temperature, is only true for ideal gases (which can neither be liquefied, nor be solidified). In fact, even in the simplest approximation to a real gas, van der Waals equation of state, there is a volume dependence of the internal energy in addition to temperature dependence. The main reason behind this is because of a potential energy associated with the gas molecules which is neglected in the kinetic gas theory for ideal gases.
Melting of ice
Your specific example of fusion/melting of ice, is a repercussion of the fact that the potential energy between molecules increases when the physical state changes from ice to water. This increase in the internal energy is provided by the external heat you provided, which is appropriately called the latent heat of fusion (latent because we do not notice any temperature change while we're providing the heat). The typical heat versus temperature graph of water looks like this:

Image source
As you can see, there are two regions where supplying heat doesn't result in any temperature change. These both regions correspond to the melting of ice and vaporization of water. Thus in general, temperature isn't the only factor to judge the amount of internal energy in a body. There are other factors like potential energy, where the given heat might be utilised without causing any apparent change in temperature, despite causing a change in the internal energy.
Work done during phase transition
Also, since the densities of all the three physical states are different, thus there's a volume change during both the processes, fusion and vaporization. Therefore, we also have to consider the work done by the system on the surroundings which is given by (in the case of isobaric conditions, which are most common in phase transitions):
$$\mathrm d W=P_{\text{ext}}\:\mathrm dV$$
Thus, according to the first law of thermodynamics, the final equation becomes
\begin{align}
\mathrm d Q=&\mathrm d U+\mathrm d W\\
\mathrm d Q=&\mathrm d U+P_{\text{ext}}\:\mathrm d V
\end{align}
Remember, that in this case, $\mathrm d U\neq nC_v \mathrm dT$, since the internal energy change is not due to the temperature change (or equivalently kinetic energy change), rather it's due to the potential energy change (which has nothing to do with kinetic energy change or temperature change).
A: Energy is required to break the molecules loose from each other in the melting process. Its called the "heat of fusion".
