Consider every "degree of freedom" of every molecule in the system as an independent place where you can store energy.
If I have $N$ molecules with $m$ degrees of freedom, that gives me $N\times m$ places to store energy. "Temperature" relates to "how much energy is there in one of these states", because by equipartition$^*$, every state will on average have the same amount of energy.
Now double the number of molecules to $2N$ - I now have $2\times N \times m$ places to store energy, and clearly I need twice as much energy to heat the system to the same temperature.
But if instead each of my $N$ molecules had $2m$ degrees of freedom, I would still have $2\times N \times m$ , and I would need the same amount of energy to heat the system that had twice as many molecules with half the number of degrees of freedom.
The situation is more complex when you have liquids - the energy is no longer stored simply in the motion of the molecules, but also in their interactions; in particular, intermolecular forces come into play, and depending on the structure of the molecules this can really change the heat capacity. A particular example of this is water, with a very high heat capacity due to the hydrogen bonding between its molecules: the energy needed to break these bonds is relatively large and dominates the heat capacity of water.
$^*$ When two systems are in thermal contact, they will tend to the same temperature; energy will transfer from the high temperature state to the lower temperature state. If you have two different ways that a system can contain energy (e.g. horizontal motion; vertical motion) then these will exchange energy until they are at the same temperature. Now if you add a third way (degree of freedom), you will exchange energy until all three are at the same temperature; and so, with more degrees of freedom, you can store more energy for the same temperature. This is one way to state the equipartition principle.