You can gain a deeper understanding of the $n-1$ term through geometry alone, not just why it's not $n$ but why it takes exactly this form, but you may first need to build up your intuition cope with $n$-dimensional geometry. From there, however, it's a small step to a deeper understanding of degrees of freedom in linear models (i.e. model df & residual df). I think there's little doubt that Fisher thought this way. Here's a book that builds it up gradually:
Saville DJ, Wood GR. Statistical methods: the geometric approach. 3rd edition. New York: Springer-Verlag; 1991. 560 pages. 9780387975177
(Yes, 560 pages. I did say gradually.)
