Question: Because, physically, if the volume is constant and the pressure increases, won't it mean that there is an increase in that internal energy?

If not, how else should we perceive it?

Answer: 
It’s called an isochoric (constant volume) heat transfer process. For an ideal gas the ratio of pressure to temperature is constant and the change in internal energy is $C\_{V}\Delta T$.

Hope this helps
For a single phase fluid of constant composition, the change in molar internal energy can be determined for any arbitrary variation of T vs P by integrating the equation $$dU=C\_vdT-\left[P-T\left(\frac{\partial P}{\partial T}\right)\_V\right]\left(\frac{\partial V}{\partial P}\right)\_TdP$$This integral will depend only on the two end states, and not on the chosen variation in T vs P used to evaluate the change. You can see that the second term is definitely not VdP. The integrand in the second term depends on the equation of state for the fluid. For an ideal gas, the integrand is zero for all variations of T vs P.