TLDR: Well, looks like I wrote a pretty long answer! I don't really see how I can summarize it in a single sentence, sorry... You'll just have to read the whole thing. :-)
You are worried about being the bearer of bad news and upsetting your friend, which is very noble, but it's not clear to me that that's in fact the right way to view the situation. From your description it sounds like both you and your friend made notable and unique contributions: he discovered the result and was the first to prove it, and you found a much simpler proof (something that's often not hard to do when you know what you're trying to prove but aren't encumbered by the baggage of all the dead ends the original discoverer of the result had to travel to get to his complicated proof). Your achievement does not necessarily devalue his, and might actually increase its value, in which case he should be happy to hear the news. Let's examine the possibilities for how your discovery can affect your friend:
Maybe your new proof is so simple that it completely trivializes the result, making it unpublishable or at best a lemma that can only be published as part of a larger paper with additional results.
Well, in that case your friend would be right to be disappointed that the result he thought he so cleverly proved turned out to be trivial. This is the only possibility I can think of where your discovery would really be bad news.
Alternatively, your new proof might be such that it makes the result seem even cooler, since while the result was and remains publishable either way, the original proof was clumsy and inelegant, which would have made for a long and uninteresting paper that not many people would enjoy reading, whereas the new proof opens up a new perspective or adds a new kind of argument that makes the result itself seem more interesting and would make for a much more interesting and elegant paper.
In that case your friend should be happy, except that:
Your friend might worry that your discovery would entitle you to be a coauthor of the paper. Perhaps he was looking forward to publishing his own solely authored paper, where he would completely "own" everything.
Well, it makes sense that the idea of having to add you as a coauthor might come as a bit of a shock to him and take some getting used to. However, if he were more experienced he would realize that having a coauthor has many advantages; the value of having written a paper with a coauthor is not half of the value of a solely authored paper -- it is more than that, since the combined value of both coauthors' contributions often makes for a paper that's worth much more than the sum of its parts. This could very well be the case here, as I noted above.
Taking this into account, in this scenario your friend should still be happy, although it might not be obvious to him that that's the case, and it's an interesting question how to make him see that. If he were to consult with his advisor, I'm sure that would help.
Finally, there is the delicate matter of your friend's ego. It may be that while from a professional point of view the news you would deliver him about the simplification of his proof is good, it would still hurt his ego and make him feel like he is stupid or inadequate for not having found the simplified proof himself.
This is another example where having some experience can help soften the blow. The truth is, in math each of us has some unique skills and abilities, and one person's skills and abilities often complement those of another person such that the second person is able to see things that the first person didn't, even if they've been thinking about the problem for much longer. This works both ways, and many of us have had experiences where others have found simplifications of our proofs or points of view, and conversely we have done the same to others. So, over time we learn that being outdone by someone doesn't mean we're stupid and the other person is a genius, or even more clever than we are. Another way of saying this is that comparing mathematicians by mathematical ability isn't a linear order relation: you cannot order all mathematicians in a line such that if A stands to the right of B then A is a better mathematician than B. I hope your friend will be able to see that. (It may be a bad idea for you to try to explain it to him yourself though; that might come across as condescending.)
Now, to conclude let me talk about the authorship of the paper, since I keep referring to you as a coauthor. The reason is that it seems clear to me that with the simplification you found you have earned authorship of the paper. Again, it's very noble of you to dismiss this as unnecessary, but you won't actually make your friend feel better by giving up authorship, since if you do that he would only feel like a fraud who writes a paper with someone else's proof. A joint paper would give the credit where it belongs: with both of you, each having made an important contribution without which the paper could not exist in the form it does -- this is something that it's crucial that your friend should understand if he is to feel good about the whole story. Second, although it's possible that you are an ego-less, yoda-like person who doesn't have trivial human desires like wanting to get credit for some cool math you did, I would dare speculate that you not only deserve such credit but will actually feel good about being made a coauthor of the paper. The fact that you don't need it for your professional success is both irrelevant, and, in my opinion, false (even if you don't work in academia, people are very impressed by things like authorship of math papers; trust me). So my suggestion is, let's save this kind of modesty for those 95% of situations in life where it is warranted and helpful.
