Burke's two essays (intended to be read together, with "On Taste" working as an introduction to On the Sublime and the Beautiful") are just kind of "meh." That is, they're only so-so as a work of philosophy, but! If I remember my German Idealism class correctly, this collection by Burke has been influential out of proportion to its actual philosophical value. It made its way across to Germany, where it was read and had a major influence upon Kant, and so affected the rest of the German Idealists as well. Hegel, Nietzsche, and the others all to some extent absorb and repeat Burke's ideas. From the Idealists, these ideas in turn influenced the American Transcendentalists. At the end of the day, despite being inferior to the more famous Reflections on the Revolution in France, On the Sublime and the Beautiful may very well have more indirect influence around the world.
First, in "On Taste" Burke argues that there is a human characteristic which is affected by art. For lack of a better term, he labels this characteristic "taste." Taste involves three human traits, the sense, the imagination, and judgment. The senses are the means by which we directly encounter art-- we see it, hear it, touch it, etc. (Well, not really "etc"- there are only two more.) Once art has passed through our senses, it encounters our imagination. This may be, according to Burke, our most important faculty. It is the place where we filter and assemble these data provided by our senses into ideas. When we combine these ideas with passion or action and impose them back on the world, we are exercising the faculty of judgment. These last two are shaped by culture, tradition, and reason, and can be exercised well or badly, depending on how our characters have been shaped. People with poor taste will exercise their imagination and judgment poorly, embrace the ugly, and ultimately make the world a worse place (an idea Burke would later take up in his Reflections). People with refined taste will exercise their imagination and judgment well, embrace the beautiful, and advance civilization.
This leads into the main event, A Philosophical Inquiry into the origin of our ideas of the Sublime and Beautiful with several other additions. (Why so long a title? This book was published in the days before dust jackets, which meant that you had to cram all the info you could onto the cover.) In this work, Burke holds that the two inspirations for art that work on our tastes (see above) are the "sublime" and the "beautiful."
The "sublime" is that which inspires fear, dread, pain, awe, and other such emotions. It is the thing that we think might be hiding in a dark corner, or the sense of smallness we get when we walk into a cathedral, or the thought of the pain that comes from being injured. Burke didn't use the example, but this is the kind of effect that a good horror movie should have on us. We should feel small, limited, and occasionally even frightened.
On the other hand, "beauty" is that which inspires in us love and devotion. Not the destructive, Romeo and Juliet kind of love, but the love of a husband for his wife, a farmer for his fields, or a citizen for his city, and so on. Beauty is that which generates the small love and devotion that drive day-to-day life and make possible human relationships on a small scale. This leads Burke to some conclusions that maybe haven't stood the test of time quite as well- including the ideas that gradual transitions are always more beautiful than sharp changes; soft colors are more beautiful than harsh ones, and so forth. In other words, he uses the things he loved as illustrations. Which isn't necessarily wrong per se, it just means that some of the particulars of his philosophy may not have aged as well as his general claims.
Yet, despite these perhaps less-than useful details, I think Burke may be onto something worthy of more attention (which presumably he gets from the aforementioned German philosophers, many of whom I have not read). It is important to note that Burke does not assume that the sublime and the beautiful are different things. In fact, they may very well be the two necessary components of great art. The greatest creative works are those which both fill us with a sense of awe and wonder in something bigger than ourselves and inspire us to a love for and delight in the mundane. (Thus, Lord of the Rings may very well be the embodiment of the Burkean aesthetic.) I don't have the aesthetic wherewithal to worth through all of the implications of this idea, but I certainly think it's an interesting one to kick around...
Over all, this work is... adequate. There certainly are interesting ideas to be found, but the final result is something a bit underwhelming. So if you're looking for a place to begin with Burke, I wouldn't recommend this text-- fortunately, the Reflections (linked above) is much better and much more worthy of your time and attention.