How do professors get to apply and work on industry projects?
Reputation, personal contacts, and open calls for proposals. 

Academics who publish important papers relevant to industry problems are likely to draw the attention of people in those industries who may reach out to the professor to find out more about their research. These kinds of contacts may turn into invitations to come speak, collaborate, or even a contract project to get paid to do research for the company.

Similarly, not everyone who gets a PhD goes into academia, so it's likely that a professor will have former labmates from grad school or postdocing or recent advisees who have gone into industry. These connections, if maintained, can be invaluable for finding out about the needs of industry and may lead to funded research projects.

Finally, some companies run research programs more or less just like government research programs. If you want to work with Microsoft Research or Google or Pfizer, you may simply go to their website, discover what research funding programs they are funding, and apply.

There are certainly other possibilities (incubators, university-level liaison offices, etc), but these three are among the most common.