Sorry if the title of this question is not formatted in a way that stackexchange likes, though my question is actually much more specific.
I was reading the book A Course in Metric Geometry, by Burago, Burago, and Ivanov. So I understand the basic idea behind length spaces and length structures seems to be an abstraction of distance. That is, that different metrics may exist on some space and we can group a subset of these metrics intro a length structure. Then a length space applies a metric over the set of metrics in the length structure. Please correct me if I am getting anything wrong.
However the book does not really explain what the mathematical motivation behind metric geometry is? Like what set of problems or questions was metric geometry designed to answer?
I am looking for some sort of statement like Tao gives on measure theory, where he describes the origin or measure theory in trying to understand the area or volume of a set. And then the development Lebesque measure as arising from the need to develop a definition of measure that went beyond simple elementary sets, as in the Riemann or Darboux integral.
So did metric geometry arise out of problems in measure theory, or from problems arising out of differential topology and riemann geometry, etc?