Advisor transferring to better school

The other school is hiring your adviser because they think they'll be a good researcher and faculty member. This means that at some level they trust your adviser's judgement.

It's not impossible, but it'd be highly unlikely that they would refuse to admit you against your adviser's request. The more likely situation, if they're concerned about your ability, is that they would ask to have you repeat some courses or milestones like an oral exam or portfolio review.

When they hire your adviser they're essentially saying that they're impressed with their research program. It'd be a little self-defeating to hire that person but say that they can't bring substantial portions (the students) of that effort.

Go with him.

Seriously heed this: Go with him

Reasoning:

You seem to want to go. The adviser wants you to come with. Gird you loins and make it happen. Put in the work.

No, it is not normally automatic* that a university will admit existing graduate students but it is probable that your professor can pull strings to help you get on board. (*for the few schools I'm aware of, anyway. For the official answer, you need to read up on admissions rules for the target university.)

Do not be an orphan student. This is very frustrating and lonely. If you do not choose to follow, find a new adviser.

(I speak from personal experience. I should have followed.)

If your adviser wants you, it is unlikely that the graduate hiring committee at the new school would get in his way, particularly if you have your own fellowship and have already published.

Another reason to go is to have continuity of your PhD, i.e. the same adviser and the same thesis.

Then, it is a stronger school. You might forge collaborations with highly successful researchers (if you don't spend too much being in awe of their accomplishments). You can learn a lot from those people. Also, a PhD from there improves your chances of getting a good job after graduating.

Your argument against is that you may not be good enough for that place. Top places can be highly competitive and there are brilliant people who work there. The real problem is not how good you are, but how well you adapt to the academic environment there. For that, you need to do the standard things: communicate with your colleagues, ask questions, ask for help, work hard, and try to ignore the feeling that the school is too good for you.