Let $M$ be a monoid and $BM$ be its classifying space. There is a model for $BM$ based on labelled configuration spaces of the line $[0,1]$. Points of the configurations are labelled by elements of $m$. The topology is such that points can fall off or move in from the boundary. If points meet, then they come together and the labels multiply. The map $M \rightarrow \Omega BM$ takes an element of $m$ and maps it to the loop that introduces a configuration point labelled by $m$ at $0$ and moves it along the interval until it falls off at $1$. This is actually a map of monoids (cf. https://twoplusonet.wordpress.com/2013/04/24/notes-on-the-group-completion-theorem/).
Why we have such a model? Is there any proof or references?