The robot manufacturers or the industries that use them will sponsor these competitions; these sponsorships generally take the form of donated equipment.
Beyond the tax writeoff of donating the equipment, a robot arm manufacturer gets the university to turn out students that have experience with their particular arm. Similar benefit for industries.
If you are, say, an automotive company, then maybe you'll donate the same kind of robot arms and PLCs that you use to a university. Everyone in the engineering department at that university will then train up on your company's preferred robot arms, using your company's preferred PLCs, and then you can come hire up the best-performing students from the graduating classes.
The company gets good PR for sponsoring the competition, a tax writeoff for donating the equipment, and gets a steady stream of people that already have a working knowledge of interfacing to the equipment they'll use in their professional career.
A lot of companies will do this, to get people used to using their software, etc. My university had everyone using Microsoft Office (Word vs. LaTeX), Matlab (vs. Mathematica), Autodesk Inventor (vs. Solidworks) and so on. Those were all provided free to me while I was a student.