That's a very complicated question, but an obvious reading is to say it has a lot to do with class. If you come from a wealthy, "old-money" family - you can rest assured than when you come to want a job, your parents or any number of their friends are going to have well-paid positions waiting for you.
So after an incredibly expensive private education, getting into an Oxbridge school isn't going to be too hard. Where students from more modest backgrounds may choose to study something with more clearly defined career paths, an upper-class kid can pick something like Philosophy, thinking it's a bit of a doss subject (perhaps). That's not to throw shade at Philosophy, but subjects like that cost just the same as others which have far more tangible job prospects.
As finding well-paid, executive work is never a concern for kids of the "elite", they can afford to read subjects that for others would simply be impractical.
Again, not to tar everyone with the same brush, but if you read about some of the hijinks famous Oxford students got up to during their time there (like the infamous Bullingdon Club), its a wonder they left with anything above a 2:2.
So how does this play into how Oxford Uni influences British politics? Well there are a couple of things you could observe from Oxford that are reflected in the British political establishment.
Debating is a big part of what the Oxford Union is about, and they frequently have famous speakers come in to debate a variety of topics (a lot of them are posted to YouTube). The despatch boxes, and often fiery exchanges - moderated by a chairman to whom the speakers should direct their sentences ("It seems to me, Mr Chairman, that blah blah..."). All of this is mirrored in the House of Commons, the house of elected Members of Parliament. So students at Oxford (if they so chose) spend time getting used to (and perhaps enjoying) this environment, and so going into politics seems appealing.
Going back to class, as a lot of kids at Oxford come from money, they are likely to have a vested interest to protect that money, and so either support or are members of the Conservative party. Conservatives (also called Torries) generally believe in lower taxes and a reduced wellfare state (though the actual strength of the left/right wing balance is far less clear than it is with Dems/Reps).
Coming from money, it's much easier then to get into politics. While MPs are paid a decent salary (and that's before perks), you may have to spend several years campaigning before you can get elected - and of course there's no guarantee that'll happen. Your average Joe needs to be getting paid all of the time to support himself.
I could go on for some time on this, but the truth is there is no real cause and effect here. Just generalizations and things we've observed over the years. A lot of leading British politicians went to Oxford, and shared memberships of various exclusive clubs and unions. The "old boys network" is clearly observable, especially within the Tory party.
So while Oxford as in institution isn't doing much to directly influence British politics (in my opinion), a large portion of kids who go there thrive (for the better and worse) in the environment and culture that exists there. This applies especially to the Torries, but can also be seen other leading parties. As the Conservatives are the current governing party, the influence of the time they spent at uni (along with the friends they made) perhaps seems more prevalent.