Read the HuffPost article here
Obama knew that success means that he could emerge the figure who will lead the world in combating terrorism, as well as how improve on how America is viewed from abroad (by other Western nations). Failure will see America condemned as a belligerent power with Obama as a reckless leader.
At the end, Obama chose this type of strategy because the rewards were very much worth the risk. It has been said before that past Presidents, particularly Bill Clinton (and possibly both Bushes), has had the opportunity to take down Bin Laden but never did. Obama knew that Bin Laden represented terrorism and is still feared by so many around the world (including Americans). This fueled his determination to take him down and made him take the risk he took.