Heat of hydrogenation of 1,3-substituted cyclohexanes
First of all, as Jan pointed out on the comment, better nomenclature for thes compounds is syn- and anti- instead of cis- and trans-, respectively. However, OP is more familiar with cis- and trans- nomenclature I'd continue it with my answer.

Each substituted cyclohexane ring (e.g., di-substituted ones here) has two chair conformers as depicted in OP's bottom diagram of methylcyclohexane (a mono-substituted one). In one conformer, the methyl group is in equatorial position (left hand structure) and the methyl group in the other one is in axial position (right hand structure). Because of two 1,3-interactions cost additional approximately $2 \times \pu{3.8 kJ/mol}$ steric strain energy, the predominant conformer is equatorial one.

Now, if you put tert-butyl group in $\ce{C3}$-equotorial position on equatorial-methylcyclohexane (left hand structure, vide supra), you get cis-1,3-substituted compound (C). On the other hand, if you put tert-butyl group in $\ce{C3}$-equatorial position of axial-methylcyclohexane (right hand structure), you get trans-1,3-substituted compound (D). Because it is too bulky, the tert-butyl group is always in equotorial position in cyclohexane ring (1,3-interaction with one hydrogen would cost additional approximately $\pu{11.4 kJ/mol}$ steric strain energy), as depicted in following diagram (The larger the substituent, the more the equatorial substituted conformer will be favored):



The most predominant cis-isomer (C) has both substituents in equatorial-positions (the left hand side structure of (C)). Other conformer (get by the ring flip; the right hand side structure of (C)) has both substituents in axial-positions. Thus, as a rule of thumb, the conformer with both substituents in equatorial positions is more stable.  

On the other hand, the both trans-isomers of (D) have one substituent in an equatorial-position and the other in an axial-position. Yet, the conformer with the tert-butyl group in the equatorial-position (left hand side structure of (D)) is more stable between two, because tert-butyl group is comparatively much larger than methyl group. However, because most predominant cis-isomer (C) has both groups in the equatorial-positions, it is relatively more stable than (D), which has methyl on axial-position, making additional steric strain energy.