Andre Gramain's 1992 exposition of van Kampen's statement is in fact covered explicitly in "Topology and Groupoids", as it was in the 1988 version of that book; it was just an exercise in the 1968 edition.
The point of this type of exposition is to say that sometimes a group can be better explicitly described in terms of groupoids. A basic example is the group $\mathbb Z$ of integers! This is obtained from the groupoid $\mathcal I$ which has two objects 0,1 and exactly one arrow between them by identifying 0 and 1. This is rather analogous to the way the circle is obtained from the unit interval $[0,1]$ by identifying 0 and 1 !
Another aspect is that sometimes a groupoid is a better object to deal with than a group. For example, a homotopy colimit of a diagram of groups is really a groupoid. This is analogous in topology to taking double mapping cylinders rather than a pushout of maps of CW-complexes.
For more information on the book see
I should also say the Higgins' monograph has results on groups, for example a generalisation of Grusko's theorem, that has not been equalled by other methods.
Another aspect of Topology and groupoids is Chapter 11 on "Orbit spaces, orbit groupoids", which allows some computation of the fundamental groupoid and hence group of an orbit space.