Why can't the LHC detect heavy particles?

It's not detecting the particles that is hard, it's making them in the collisions.

Although the LHC collision energy is 14TeV, collisions aren't between the protons but rather between individual quarks inside the protons. Since the energy is shared between the three quarks in a proton the actual quark-quark collision energy is a lot less than 14TeV. Even then, for various reasons to do with conservation of momentum not all that energy can go into creating new particles. The end result is that it's hard to create particles much about above a TeV in weight. More on this in What is the maximal particle mass one can create via the LHC? Can we create dark-matter particles via the LHC? if you want to pursue this further.

The upshot is that if the heavy particles have a mass much greater than a TeV the LHC can't create them, and obviously if they can't be created they can't be detected. All is not completely lost since we might be able to detect heavy particles indirectly by the influence they have on the collisions we can detect. Even so the upper mass limit is still restricted.

As a consequence of $E=mc^2$, to create a heavy particle (i.e. large $m$) requires a large amount of energy ($E$). Since the LHC only generates a finite amount of energy in the collisions, there may be particles that are too heavy to be produced. This looks likely to be the case for superpartners (if they exist).