Why do particle accelerators need to smash particles together? In a particle accelerator particles gain a large kinetic energy before being smashed into each other. After the collision the kinetic energy goes into the rest mass of new particles.
Why doesn’t the kinetic energy produce new particles before the collision?
I assume that it is because there is an inertial frame moving with the particles in which they don’t have any more energy than their rest mass.
Is that correct?
 A: 
Why doesn’t the kinetic energy produce new particles before the collision?


I assume that it is because there is an inertial frame moving with the particles in which they don’t have any more energy than their rest mass.

A proton in the LHC beam has high energy, 6.5 TeV, , and yes in its center of mass has zero kinetic energy, but the reason it does not break into more particles is because there is no lower mass bound state of quarks and gluons to which it could decay. A muon of such an energy, for example would have a probability to decay to an electron and an electron antineutrino  before colliding with the opposing beam ( supposing one could have a muon muon collider). If the proton could decay, the total energy would be distributed according to the quantum mechanical probability to the decay products, not just the kinetic,
To come to the title

Why do particle accelerators need to smash particles together?

They need to smash particles together in order to study the interaction/scattering behavior at high energy.
A: We use particle accelerators to put different kinds of particles close enough together for them to experience interactions, which we can then detect. The easiest way to do this difficult job is to turn the particles into tiny bullets and shoot them at the desired target at high energies. This is the particle accelerator's job.
For example, Rutherford's original experiments scattering alpha particles off of gold atoms used alpha particles (from a radioactive decay source) which had enough energy to penetrate the electron cloud surrounding the gold nuclei and instead of scattering off the electron cloud, went on "into" the gold atom to ricochet off the nucleus instead. Nowadays we can use a particle accelerator instead of a radioactive source to do the same type of experiment, and with the right design the accelerator can produce an enormously more intense source of "bullets" than a radioactive source.
Now, if we want to look deeper still into the nucleus- and try to "look" inside a proton and see what's in there, for example- we need the bullets to be made so energetic that their de Broglie wavelength is way shorter than the dimensions of a proton. When this condition is met, as it was in the SLAC accelerator in the late 1960's, you can use extremely energetic electrons as the bullets and thereby  resolve the insides of that target proton and detect the presence of three quarks running around in there.
