The error in the image you posted indicates that there were problems during the download of Revit 2013 executable and the install content cannot be successfully extracted. The only solution for this problem is to download Revit 2013 executable again and extract the install content from the new copy.
Once you get past the install content extraction problem, I believe I know what is causing your installing to hand (aka take an extremely long time to complete). Based on your screen capture, it appears that you are using a non-English operating system. If this is the case, then the Revit install automatically enable each content pack, which means the install has to download Revit content for each available language and this can be a lengthy process. My recommendation would be to expand the Autodesk Revit 2013 entry on the Configure Installation page of the install, scroll down to the content pack area and disable the checkboxes associated with country you don't need content for. This should reduce the amount of time required for the install to complete.