Looking at the World Universities Ranking for 2018, Oxford comes in 1st, Cambridge 2nd, Imperial 8th, UCL 16th, LSE 25th, Edinburg 27th, KCL 36th (I have just looked at top 50) so I guess the ranking is part of its appeal, as well as its relative affordability (rents are cheaper than many other places such as Durham and London universities), there are often generous grants for students from low income backgrounds, both cities are considered attractive and are walkable, they are also close to London and have higher contact time through the tutorial system. I hope that answers the why.
How hard it is to get in is a tricky one - it depends on the course. In broad terms, you need very good GCSEs and predicted A Levels, to do well in entrance tests (where applicable), to perform well enough at interview, sometimes to submit school essays. I think the stamina and consistency that this requires make it more difficult than most other unis. That said, Medicine anywhere, seems to be tough.
That said, we are very lucky in the UK to have lots of excellent unis which rival Oxbridge/are better/more suitable/cater for certain subjects, especially for STEM ones.
If you were in USA, you would find a parallel situation re Ivy Leagues among parents and young people. What I like about the UK is that people don't automatically go to their local uni. My nephew is at uni in France and goes to his local one in a large provincial town and although it costs next to nothing, the education he is getting is not good.
May I ask what your home country is?