Every program handles admissions slightly differently, sometimes even within the same university.
Some allow faculty to select their own students, with very light oversight from the rest of the university.
However, other programs make admissions decisions via a committee, which chooses a cohort of students based on the applicants' qualifications and the program's needs and capabilities. For example, the committee might want to ensure that wet and dry-lab approaches are both represented in the cohort, perhaps with a skew that matches future TA needs. This is particularly common in programs where one does "rotations" and formally chooses an advisor in the 2nd or 3rd year.
In these departments, no individual professor can unilaterally offer a place to someone, no matter how good they may be. However, professors can advocate for their preferred candidates in these departments. For example, if you were ranked in the middle of the pack overall, the prof might argue that you should nevertheless be admitted because you're a perfect fit for his particular lab.
It sounds like the professor is offering to do this for you. This does not necessarily guarantee you a spot--other professors might make even more compelling cases for their own preferred applicants--but it is certainly a positive sign. It is therefore important that you mention this professor by name somewhere in your personal statement. Personal statements often conclude with a section describing how your interests and skills mesh with the program's focus, and this would be a natural place to do so. Likewise, follow up by email once you've submitted your application so he can bring your application to the committee's attention.
You should, of course, continue to submit applications elsewhere too. Good luck!