The differences between undergraduate and graduate texts seem to me to be no more dramatic than those between high school and undergraduate texts, and I don't consider it especially problematic. But there are some differences:
Why are graduate texts written that way? Why isn't there a plethora of classic texts at the graduate level?
There certainly is, but there are also plenty of mediocre books. There's nothing special about graduate texts in this respect. Classic examples of anything will always be rare and noteworthy, more or less by definition.
You should keep in mind that graduate texts cover an enormous breadth of material (far beyond the undergraduate curriculum), and the upper limits of their depth can be impressive. Some books are written for very advanced courses aimed at senior graduate students specializing in a certain area. Of course they won't be easily accessible, since the research community barely understands this material. With luck, fifty years from now we'll have clearer expositions. In the meantime, we should be grateful that the author at least managed to distill the research literature into a coherent textbook, even if it is not yet easy to read.
Other graduate texts are intended for introductory grad courses. Certainly there are some bad books out there, but in my experience there are plenty of beautiful introductions that are accessible to students with a strong undergraduate background. It may still take more effort to read them than a typical undergraduate textbook, the same way an advanced undergraduate book may be tougher than one aimed at first-year students, but that's not a bad thing. The increased difficulty comes with corresponding rewards.
You can also find especially accessible textbooks. For example, some graduate texts are written specifically to ease the transition to graduate school for students with weaker backgrounds. They aren't necessarily advertised that way (to avoid putting off readers), but some books are pretty widely known to be less demanding than others. They can be a great way to fill in any missing background or just get used to graduate texts.
And it's worth having many different texts available for the same material. Undergraduate courses can sometimes be fairly regimented, with clear expectations regarding exactly what students should already know and what they need to learn now. Graduate school is much more diverse. One book may be entirely appropriate for someone with certain preparation, but utterly inappropriate for others.
If there's a subject you'd like to learn about, the best approach is often to ask for reading advice from someone who knows your background and tastes. If you can't get personalized advice, you can always take a look at lists like this. Once you have some candidate books in mind, it's worth going to the library and gathering copies of them as well as any similar books that catch your eye on the shelf. Then you can spend an hour or two comparing them to see which might best serve your needs. It's going to take you far longer to actually master the contents, so a couple of hours is nothing if it helps you find the right book for you.