I'm not a banker, so the following is only my idea of how it works:
My understanding is that exchange rates are set by the market, ie each currency finds its price by how much in demand it is. That's affected by interest rates, and also by market sentiment. In general, the lower interest rates are, the less attractive a currency is to hold so people sell it and the price drops. Governments can affect that by buying/selling their own currency as the UK tried when it was trying to stay in the ERM
In the case of sterling, low interest rates mean a low pound, but there is also not great confidence in the future performance of the UK economy. Hence a weak pound. The dollar of course is in a similar situation, but the US is big and the dollar is a reserve currency so it tends to perform a bit better all other things being equal.
2 dollars to the pound was too high anyway; it made it very hard for UK companies to sell into dollar markets and there was areal disparity in buying power between the two countries. 1.5 isn't overly low on the scale of things, though 1.6-1.7 might be a more equitable level.