This is probably a wrong answer (as I am in high school) but this is how I interpreted it. - for n=1, the vector equals 0 as stated, and the vector space is thus in n-1 (zero) dimensions or just a dot, -for n=2, there are now 2 vectors, v1+v2=0. When you add vectors you add them head to tail. Because the added vectors equals 0, v2 has to "counteract" v1 by starting at v1's head and ending at v1s tail. This creates a straight line, or the 1st dimension and fullfills the rule(n-1=2-1=1) -for n=3 there are 3 vectors v1+v2+v3=0. This creates a triangle that ends up where it started, and thus equals 0. A triangle is is flat, and thus in the second dimension (n-1=3-1=2). You could probably apply this to higher dimensions, idk if you need a mathmatical proof of it also sorry for mobile format.