How does a particle accelerator make new elements?

Its not out of nothing. They smash elements together at a large fraction of the speed of light. This allows chemicals to bond and rearrange synthetically and create particles not typically capable of occuring in the natural universe.

The difference between one element and another is the number of protons in the nucleus.  The particle accelerator can make new elements by forcing more protons into the nucleus.

It is not just a matter of firing a proton at the nucleus, however.  Protons repel one another naturally, so to hold them together requires more neutrons in the nucleus as well.  Still, the simple answer is to bombard the nucleus with subatomic particles which are moving with enough energy to enter the nucleus but without enough energy to blow the nucleus apart (or to knock one or more particles out of it).

Those particle accelerators are operating at much lower energies than modern accelerators can achieve.  Modern ones actually make individual subatomic particles hit one another so hard, the subatomic particle itself is blown apart and new quarks are sometimes discovered.

"This allows chemicals to bond and rearrange synthetically..."   Not really.  The changes that take place are within the nuclei of the target atoms.  There are no chemicals that are bonding.  That is for chemical reactions. "Bonding" involves the interaction of electrons.  The making of new elements involves nuclear reactions, not chemical reactions.

Very simply, Imagine a gun that shoots only protons (the particle accelerator).  When an accelerated proton hits the nucleus of a target atom at a very high speed (near the speed of light), and it "sticks" in the nucleus, then the atomic number of the target atoms is increased by 1 and a new element is produced.

And lastly, the new elements are not made out of nothing.  You must start with target atoms and fire charged particles at them. All the particles involved are quite tangible.