UCLA undergrad here. I'm currently working on my masters, so I've taken a number of grad classes and I think I've got a pretty good idea of what the grad program is like, as well as the culture of the department, and certainly the campus as a whole.
There are a number of younger professors and very active researchers in the department, so you do get the feeling that it's a happening place to be if you're looking for good research. This is certainly the case for the analysis & PDE group, which attract a lot of the entering grad student's attentions. Most of the professors I've met are very nice, normal people, and while some are a little "aloof", I wouldn't say any are entirely unapproachable. In terms of general atmosphere, I would say that there isn't much competitiveness among the grad students I've met. The atmosphere is, by and by, friendly and quite chilled. Also, most of the professors are really great lecturers as well, which is a big plus in my books.
Compared with other schools, UCLA's math department is relatively large. There is a lot of diversity among the grad students in terms of interests (mathematical and otherwise), and so I wouldn't say that the typical stereotypes about grad students in the sciences necessarily applies. The course-load is about 3 classes per quarter, which can be a good deal of work. TAing and such depend on fellowships, funding, etc. The "initiatory qualifying exam", also called the Basic, is not difficult, and taking the summer boot-camp course is more than sufficient preparation. However, the Quals themselves are quite tough (specifically the analysis Qual), so people take them very seriously here. How much free time you have, and what you choose to do with it, is really up to you. I know some grad students that do research 24/7, and others that manage to pursue a number of other interests. The department works you hard, but it isn't trying to kill you.
Finally, speaking in more general terms, the mathematics dept. itself is a nice-enough, if a little old, building, and it occupies a few floors, so that it doesn't feel like you're confined into very narrow corridors all day. The campus itself is lovely and quite a fun place. There are a lot of amenities to take advantage of, and I know that a number of math grad students like bouldering/rock-climbing at the gym. The weather, modulo certain inconsistencies on a set of zero measure, is sunny with blue skies, and it remains fairly temperate throughout the summer and the winter, so that we don't get much fluctuation in weather. The housing itself is located in the adjacent Westwood Village, which has a number of shops and restaurants, and is generally full of college students. Public transportation isn't great, but with a bit of effort you have access to the bulk of LA.
I hope that clears some things up. Do you have any follow-up questions?