The most radical members of the Republican Party (Grand Old Party/GOP) decided that since they do not have the votes to achieve their aims and policies, they should use their power to shut down the US Govt. as a way of forcing the Obama Govt. (Democratic Party) to fulfill their demands.
It boils down to the Republican Party threatening to cause untold problems for the US economy and its people just to get their way.
In the entire shutdown drama the Obama Govt. is asking for nothing.
This is what I have gathered from various online sources, if I have missed something or am not correct on some aspect, please feel free to add a comment, and I will make a change in the post.