Let us start by remembering that light is a wave. A wave is characterised by two quantities: its wavelength and frequency. The wavelength is the spatial distance between two peaks in the wave; the frequency the time between two peaks of the wave passing a fixed point. The curious thing about any wave is that it is (roughly) insensitive to anything smaller than its wavelength. In particular, we cannot use visible light to see anything smaller than about 400 nanometres, the wavelength of purple light.
To help get a feel for this blindness to short length scales, consider a different type of wave: water waves. These are things we have all seen, and more to the point we have actually seen the wave structure (in contrast to light, which oscillates much, much faster than the human eye or brain could observe). Imagine a pole standing in the ocean floor, tall enough to breach the surface (presumably we're close to the shore here). If the pole is small, the sea waves will pass it by unchanged. If we moor a boat or large ship to this pole, however, suddenly the sea waves will change, due to the presence of something bigger than the wavelength.
Well, what about using light of a shorter wavelength? After all, visible light is just electromagnetic waves of a relatively small range of frequencies. Surely there exists microscope-like objects for other wavelengths? Well, in a sense they do. In particular, X-rays have a wavelength comparable to atomic sizes, and indeed X-ray crystallography is a common tool to study atomic structures of newly synthesized chemicals. One big difference is that we don't have lenses for X-rays, so instead of getting images we get scattering patterns. The scattering can be mathematically related to the position of the atoms, so a simple inversion lets us get the information we need.
What about sub-atomic scales? Can we continue to use light of ever shorter wavelengths? To an extent we can, but there is a key practical problem. Quantum mechanics tells us that nothing is truly a wave, nor is it truly a particle, but some combination of both. In particular, for any wave there is a smallest possible "amount" of wave you can have, which we can think of as the wave "particle". This particle is, like almost all quantum phenomena, far too small to be observable in our ordinary life. But once we get down to atomic scales, we have to care about this. And there's another problem: as Planck and Einstein realised at the beginning of the twentieth century, the energy of a particle of light increases as its wavelength decreases. It gets harder and harder to produce light of these short wavelengths for this very reason.
But! There is a solution. For light, something we normally think of as a wave, the particle (photon) energy is inversely proportional to the wavelength. French physicist de Broglie realised that the same is true for things we tend to consider particles; their wavelength is inversely proportional to their energy. So given an electron, for example, I can give it more energy and make it have a smaller wavelength. And it is much, much easier to give an electron more energy than a photon; all I have to do is put it in an electric field.
And this brings us to our answer. Particle accelerators give more energy to electrons, protons or other charged particles, which from a wave perspective means they have a shorter wavelength. This allows them to probe the short length scales we are interested in. And all the machinery becomes a giant microscope.
There's a lot more to say about collider design, but I want to finish this at some point and actually publish it. So here's a link to a PhD comics strip that covers some of what I've said, only with pictures and jokes!
 That's a subject for another post.
 If you know about wavepackets, you'll know that this is strictly only true for plane waves that fill space. Realistic waves actually have a range of wavelengths and frequencies. However, the range is in many cases narrow enough that we can treat these values as single quantities.
 Undergraduate level, involves calculus.