I had never heard of this software, so I did a bit of research. Unfortunately, the results were complicated. Specifically:
1. One website,
http://graphicburger.com/long-shadow-generator/ , says that it is a layered PSD file, not an action, nor plugin, nor extension.
2. A second website,
https://www.behance.net/gallery/10655837/Long-Shadow-Generator-Free-Photoshop-Action , unambiguously says that it is an action. (Note that actions generally will not be compatible between different versions of PS).
3. A third website,
http://www.hongkiat.com/blog/long-shadow-generator/ , says that it is indeed an extension and is installed using the extension manager, but notes that there are two versions -- one version for PS CS6, and another for PS CC.
This info strongly suggests that there are multiple versions out there, all with similar names.
Because of this, I would suggest:
a) If you have a small number of shadows to make, just make them yourself using native PS tools. It's quite easy to do. If you are interested, let us know because there was a old thread on this forum describing how to do this, and I'll be happy to hunt it down for you. If you have to cast shadows from a large number of objects, then it might be worth you while to figure out why this piece of SW won't install using the method you have tried.
b) If you want to pursue the use of this software, then you should do some research yourself to determine compatibility and appropriate installation methods. You will need to find out what version of PS you are using, whether or not you, as a student with a regular account on that computer have sufficient privileges to install software, and exactly which version of the software you are dealing with.
HTH,
Tom M