What does it mean, that a professor says: "I think it would be great to have you at X university, and your suggested dissertation topic is a great idea, But, I think you should apply broadly, not just to my university" ?
I mean, does that imply he's really into accepting you? Is there any hidden subtext here?
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.
Put yourself in the shoes of a professor trying to recruit a PhD student. You get many emails from students wanting to work with you. Even if you find one that you like a lot, the application deadline has not passed yet and there is still a chance you will find a student who’s even more impressive and more worthy of the position. (Also, it may be that it’s the department who needs to admit the student so even if you are sure you want the student, you cannot guarantee to them that they will be admitted since other people are involved in making those decisions.)
You also know that any word you write in an email to the student will be seized on by the student (and analyzed and over-analyzed to death, in conversations with friends and on academia.stackexchange) as evidence that they are going to be admitted or as evidence they will be rejected, potentially leading the student to make bad decisions in case the words are misinterpreted. So you do not want to set up unrealistic expectations. Imagine saying to a student “I was very impressed with you and think you have a great dissertation topic. Please apply!” Well, the student may well interpret this as “you’re in” and not bother applying anywhere else. This is a recipe for disaster if one of many things that can go wrong between now and the time that you receive an official acceptance letter, does. It is much better to be cautious and use language that indicates to the student that while you like them, nothing is guaranteed at this point.
Bottom line: there isn’t a subtext. “Apply broadly” is sound advice to give to anyone, both good applicants and less good ones. If anything, this is a small sign that the professor is a person of integrity who does not try to manipulate or mislead applicants to further his own self-interest. But even that is a very weak signal at best.
I think the statement is one of general encouragement. But take the words literally. They might even be boilerplate if it was a first email exchange. In the US, individual professors don't normally have the power to admit students. Elsewhere they might, but I'm guessing that if he has the power you aren't his top choice. Apply broadly.
If it is something other than boilerplate then the person seems to be saying that you are a good candidate, generally, for graduate study, but that, for whatever reason, your acceptance here isn't guaranteed. You may be trying to apply at too competitive an institution, or even one that isn't competitive enough.
I would normally just ignore a lot of blind emails from students, especially when it was clear they were in some different field and had no knowledge of what I did. But, perhaps this person is a bit more polite and, thus, boilerplate is the result.
