Why could interest rates go up if Brexit happens?
What would happen in the event of a "Leave" vote in the referendum? Well, the pound would quickly fall in value against its major trading partners - and some falls have already happened as the "Leave" vote appears to increase in probability. That makes imports more expensive, which is directly inflationary. Which pushes the Bank of England (the UK's Central Bank) to raise interest rates in response.

On top of that there are second-order effects which would pull in the other direction, but over a longer time period: there's the directly deflationary effect of the significant increase in unemployment. And the increased risk of further disintegration of the EU, with all the consequential risks to peace and prosperity that that would bring.

And as Alexis L. points out, the UK leaving the EU adds significant risk to the UK's currency; and that corresponds to a need for a higher return on the currency: that return is the interest rate. That's because if interest rates are too low, then people would keep selling the pound sterling, and it would keep dropping in value, pushing up inflation (requiring higher interest rates), and harming UK companies that rely on exports (creating political pressure to do something about the exchange rate - and that "something" is higher interest rates).