Are there intuitively clear and not technical proofs of homotopy excision theorem? The proof given in May's "A concise course in algebraic topology", for instance, is not very involved, but quite technical. Are there less technical, but more "ideologically profound" proofs allowing to see "why" it should really be true in the range of dimensions?
 A: The ncatlab entry on the "Blakers-Massey Theorem" gives more information, and references below are  to the papers cited there.  Blakers and Massey   wrote three papers on this subject, and paper III (1953) gave conditions for an algebraic determination of  the "critical group", that the map 
$$\pi_m(A,C) \otimes \pi_n(B,C) \to \pi_{m+n-1}(X;A,B)$$given by the generalised Whitehead product is an isomorphism. Note that the tensor product given is zero if one of the factors is zero; this is why the algebraic result implies the connectivity result.  So there is a case for referring to the previous two papers as giving the "Blakers-Massey connectivity theorem".
One of the conditions for their result is that  $C$ is simply connected. 
This condition is relaxed in a paper of Brown-Loday (Proc LMS 1987) by using a nonabelian tensor product announced here and developed in Brown-Loday (Topology, 1987). 
For the connectivity result, the book by Munson and Volic has a prepublication pdf available here. This could come under the heading of "intuitively clear", though it is quite a lot of work. 
The meaning of the algebraic result is to my mind "intuitively clear" and immediately believable, but the proof of the main "van Kampen theorem" in Brown-Loday Topology (1987) uses advanced methods of algebraic topology. 
The paper by Ellis-Steiner (JPAA, 1987) uses the methods of the above Brown-Loday papers to solve the old problem of the critical group for $(n+1)$-ads, for which the connectivity part is explained in the book by Munson and Volic. 
There  seem to me to be prospects of developing further the methods of the B-L Proc LMS paper. 
