I think quite a few regular readers can relate to situation you described, and it is not easy to way out. :-)
If documentation is not required by customer, and is used only for internal process (like if your customer is internal, and IT is just a cost center, or you are developing a website for a startup), people are tempted to develop as little documentation as possible to carry them to the next day, not "waste" time (on documenting something which is going to change soon anyway), and spend time to solve more urgent problems instead. Like in startup, if you solve problems for users and make them happy, you survive for another day even without proper documentation. No amount of documentation will save you if you don't deliver product. So you accumulate technological debt, layer by layer, and your progress slows.
Hard trick is to realize when you are not startup anymore, and you need to start paying down that debt (instead of adding new features to make new customer more happy). Likely company you interviewed realized that.
You start by establishing processes, and it is hard going because nobody wants to change, and old ways were good enough up until now... How you eat an elephant? One bite at a time.
There is no replacement for real life experience. That's why they asked you if you have experience doing that.
That is not an answer, but many books were written about how to do it, and I am not going to write one now.