Q:

Why do we need high energy to explore small dimensions?

I am taking a quantum physics class, and for the life of me, I can not remember why we would need a vast amount of energy to understand the microscopic universe.

A:

Frequently one probes matter by bombarding it with radiation or with other pieces of matter, and then looking at the products. This is called a scattering experiment. Since the probe system is quantum, whether or not it is made of light or matter, it is associated with a wavelength. The de Broglie relation tells you that this wavelength is $\lambda = h/p$, where $h$ is Planck's constant and $p$ is the momentum of the probe. 
This de Broglie wavelength gives a lower limit to the resolution of the probe. Any feature smaller than this will simply be smeared/averaged over the probe's wavelength, and so will not be visible. For the same reason, normal microscopes only work down to a few hundred nm (the wavelength of visible light). Electron microscopes allow you to see smaller features because the electrons have a smaller wavelength. 
Therefore, the smaller the feature you would like to see, the higher must be the momentum, and thus the energy, of your probe system. This is one (oversimplified) reason why the LHC has to be so large and achieve such high energies (relative to usual microscopic scales).

A:

Apart from the fact that high energies correspond to short wavelengths, as has been explained in the other answers, another reason for studying high energy collisions is noteworthy as well: 
Many unstable elementary particles have large masses and, due to conservation of energy, can only be produced by decay from highly energetic systems. Examples include heavy quarks, W- and Z-bosons and basically all hadrons except for protons and neutrons. So if you include knowledge of all elementary particles and their interactions in the category "understanding of the microscopic universe", then this is definitely an answer to your question.  
Below you can find a table of the elementary particles and their masses. As you can see, many of them exceed 1 GeV, which is very heavy compared to electrons, which we experience in daily life.

A:

You have probably seen the relation between energy and frequency for photons $ E = hf$. Since the speed of a photon is $c$, the wavelength is $\lambda = c/f$ so $$E = hc/\lambda. $$ Therefore short wavelengths correspond to high energies. 

