It's a combination of limited budget and the time required to film the episodes. Unlike the production-line methods used in US TV allowing for 22-25 episodes a season, most UK programming requires longer time to produce (in part due to union limits on work hours and the general attitude of quality vs. quantity). It takes more than 8 months to film an average season of Doctor Who; in order to produce more than 13 episodes the series would need to adopt a nearly year-round schedule.
There is no set rule saying seasons must be 13 episodes long. Indeed, technically most seasons are 14 episodes because production usually includes a Christmas episode. Series 7 actually counted the special as part of the regular season (normally they're considered standalone episodes), giving it a total of 14 episodes. Officially, Series 4 was 18 episodes long because the BBC considers the "gap year" specials that aired between Christmas 2008 and New Year's 2010 to be part of that series. Series 3 of Torchwood was 5 episodes long. Series 8 is only 12 episodes long, but that was in part due to its premiere episode being "feature-length" and it is not yet known whether the 2014 Christmas special will be considered Episode 13.
This is not unique to Doctor Who. Game of Thrones only airs 10 episodes per season, for example.