I'm trying to understand what paste script and paster are. The website is far from clear.
I used paster to generate pre-made layouts for projects, but I don't get the big picture. 
As far as I understand, and from the wikipedia entry, it says it's a framework for web frameworks, but that seems reductive. paster create seems to be able to create pre-made layouts for setuptools/distutils enabled packages.
What is the problem (or set of problems) it's trying to solve?
PasteScript (and its companion PasteDeploy) are tools for running Python code using 'entry points'. Basically, a python library can specify in metadata that it knows how to create a certain kind of Python project, or perform certain operations on those projects. paster is a commandline tool that looks up the appropriate code for the operation you requested. It's a very general kind of problem; if you're familiar with Ruby at all, the equivalent might be 'rake'.
In particular, PasteDeploy is a configuration format to serve Python webapps using paster. Both PasteScript and PasteDeploy are important for the Pylons web framework.
What is paste script?