Oxbridge entrance processes are some of the most competitive and rigorous, with extra entrance exams and interviews. In a way yes, it is to "show off", because getting into Oxbridge is an impressive feat, and the higher this number is the better it reflects on the academic achievement of the school. This isn't true in all cases, some less mainstream courses may be fairly easy to get into, and some popular courses in other competitive universities may be harder to get into. But overall it's because people regard Oxbridge with a certain respect (correctly or otherwise).
However, getting into Oxbridge or another top university is not just about "league tables and reputation", they have a record of high quality education and students end up well connected for their future graduate careers.
I find it strange that you compare going to a top university to "going straight into a job". First of all schools are academic institutions so it makes sense that their academic ability as a system is reflected by the number of students going into higher education. Secondly, the jobs that an 18 year old can go straight into are often not the most desired or respected professions, as in many cases these would require a degree to pursue, so it wouldn't be of interest to most prospective parents to hear about these.