How to interpret "I think you should apply broadly" from a prospective PhD supervisor?
You should interpret it as good advice.
Apply broadly. Find the program that's the best fit for you. Many graduate programs are sufficiently competitive that many qualified applicants are not admitted. It's impossible to admit everyone qualified who applies if there are more qualified applicants than slots, and there is a lot of subjectiveness and a bit of luck in who makes that cut.
Maybe the professor is gently saying you are unlikely to be admitted to their program and lab, maybe not; there is little benefit for you in attempting to divinate their meaning. There is lots of benefit for you in following their advice (more choice, higher chance of admission), and risk in failing to do so (no choice, chance of not being admitted and not having a backup plan).
Sometimes advice is just advice.