What determines how many master/PhD students a given professor can take (STEM)?
At American universities, each department is allowed to accept a certain number of PhD students each year.  This number is determined annually by a dean or provost's office, primarily based on funding and student retention.  It could also be limited by the supervisory capacity of the faculty, but funding is usually more important.  Each department has its own procedure for matching students with supervisors.  Often the procedure is for students to go and ask faculty to supervise them.  In principal, there is nothing stopping a single faculty member from taking all the graduate students.  This probably happens from time to time in small departments.  

In the case of the rumour you mention, we can guess that this professor does not have access to additional TA/RA funding at this time, and is currently only taking students with external funding.  Typically TA funding is linked to the department, while RA funding is linked to a particular faculty member's grant.