I'm pretty dumbfounded by this, considering that he never mentioned
this before and that I've never come across a situation like this.
It's not even a trust issue! It's pure feedback. Why would you keep
feedback from someone anyway?
First, let's be clear, you are not in a position to pass judgment on your professor for making you waiving your right to view the letter a condition to writing it. It is his right to make this a condition for any reason whatsoever (or even for no reason at all), and he is not required to provide any explanation. So, my main advice is that, whatever you write to him, make sure that it doesn't contain even a trace of the slightly judgmental attitude I am sensing in the question you posted here. Just say that you were not aware this would be an issue, acknowledge that you understand his request and that you have complied with it.
With that said, since you ask, there are some perfectly logical reasons why a professor would not want their student to view the letter they are writing. The most compelling reason for me would be the following: a letter, even a very positive one, contains statements about the subject of the letter that normally one would not be inclined to tell someone about themselves in person, both because it's awkward and because it is potentially counterproductive. The knowledge that the subject of the letter would not be reading it eliminates this problem, and provides me with reassurance that I can state my true opinions without fear that my writing would be affected (positively or negatively, and consciously or subconsciously) by any concerns about how the subject would receive those opinions. Again, I emphasize, this reasoning is valid for very positive letters (as well as more obviously for letters also expressing some negative opinions).
As an example, say I believed that a student was the greatest mathematical genius since Euler. I would certainly want to say that in a letter of recommendation, right? Now, you might think that I should have no reservations about telling that to the student as well. I mean, why not -- surely he/she would be flattered and feel great, right? Well, the thing is, hearing that you are the greatest genius since Euler may actually not be good for you, even if it's true. It could inflate that person's ego to the point of making them arrogant and obnoxious, and demotivate them from working hard since they would reason they can achieve great success with very little effort, and maybe have various other negative effects. So, you see, not wanting them to know this opinion that I hold, and not having to worry that whatever I write about them will be influenced by such concerns about secondary and unintended effects of my writing, are completely valid and legitimate reasons for me to request this kind of confidentiality.
The bottom line is, not waiving your right to view the letter is essentially tantamount to making a rather aggressive demand for "pure feedback" from the professor, taking advantage of the fact that he is bound by a promise to provide honest feedback about you to someone else. I think your professor is absolutely justified in refusing to cooperate. If you want feedback from him, ask him for it in the right way and at the right time, without tying your request to anything else related to external circumstances, so that he'll have the ability to consider your request without any pressure and decide if that's something he wishes to do.