The key facts used here are that the category of crossed complexes is monoidal closed, and that there is a good interval object, essentially the groupoid extended to be a crossed complex.
Notice that by other work of Brown-Higgins, crossed complexes are equivalent to strict globular omega-groupoids, and also to strict cubical omega-groupoids with connections. In fact the monoidal closed structure on crossed complexes is deduced from that on strict cubical omega-groupoids.
Ara, Dimitri; Métayer, François, The Brown-Golasiński model structure on strict ∞-groupoids revisited. Homology Homotopy Appl. 13 (2011), no. 1, 121–142.
Maltsiniotis, G. La catégorie cubique avec connexions est une catégorie test stricte. Homology, Homotopy Appl. (11) (2) (2009) 309 – 326.