TITLE: What is the intuition behind the Freudenthal suspension theorem?
QUESTION [27 upvotes]: The Freudenthal suspension theorem states in particular that the map
$$
\pi_{n+k}(S^n)\to\pi_{n+k+1}(S^{n+1})
$$
is an isomorphism for $n\geq k+2$.
My question is: What is the intuition behind the proof of the Freudenthal suspension theorem?

REPLY [3 votes]: Edit October 2013: I have given more information on the Blakers-Massey triad theorem on the ncatlab here, referring specifically to the algebraic determination of the critical,  i.e. first non vanishing, group. This is described as a tensor product of relative homotopy groups, and of course a tensor product is trivial if one of the factors is trivial. So we get an intuitive view of the connectivity result and so for the Freudenthal Suspension Theorem. Indeed, they are both seen as a consequence of a higher order homotopy Seifert-van Kampen Theorem. 
Original answer: 
John Klein's answer refers to the Blakers-Massey triad theorem; this was generalised to an $n$-ad connectivity theorem by Barratt and Whitehead, and of which Goodwillie has given a geometric proof, using general position arguments. 
In these results there is interest in describing the critical group, an $n$-ad homotopy group, which may be nonabelian in general. Some abelian cases have been described by Barratt and Whitehead, following Blakers-Massey, with simple connectivity assumptions. 
This problem was solved in Theorem 3.7 of 
Ellis, G.J. and Steiner, R. "Higher-dimensional crossed modules and the homotopy groups of $(n+1)$-ads." J. Pure Appl. Algebra 46 (2-3) (1987) 117--136.
which uses crucially the van Kampen Theorem for $n$-cubes of spaces in 
Brown, R. and Loday, J.-L. "Van Kampen theorems for diagrams of spaces", Topology 26~(3) (1987) 311--335. With an appendix by M. Zisman. 
Of course connectivity results follow from the algebraic results, but the proof in this last paper is by induction with connectivity in one dimension implying surjectivity in the next. 
One intuitive point is that the description of the critical group of an $n$-ad will involve generalised Whitehead products coming from sub $r$- and $s$-ads of the given $n$-ad, and an elaborate  algebra (crossed $n$-cubes of groups, as defined by Ellis-Steiner) is needed to handle all this. The proofs do not involve general position at all. 
For a fairly recent application of these ideas, see
Ellis, G.~J. and Mikhailov, R. "A colimit of classifying spaces". Advances in Math. (2010) arXiv: [math.GR] 0804.3581v1.