Recently I'm studing some basic homotopy theory. An important brick of the exact sequence of cofibration is the following sequence: $$X \stackrel{f}{\longrightarrow} Y \stackrel{i}{\longrightarrow} C f \stackrel{\pi}{\longrightarrow} \Sigma X \stackrel{-\Sigma f}{\longrightarrow} \Sigma Y \stackrel{-\Sigma i}{\longrightarrow} \Sigma C f \stackrel{-\Sigma \pi}{\longrightarrow} \Sigma^2 X \stackrel{\Sigma^2 f}{\longrightarrow} \cdots$$
Is there a deep interpretation of this sequence or it's just a technical tool to obtain the theorem?
Actually I do know that homotopical (co)limits and model categories should answer to my question but it's like looking for a needle in a haystack.
I do not know if it anwers your questions, maybe it is just a long comment, but most of the times (if not all) exact sequences of (co)homology and homotopy (in algebraic categories) underly related structures (up to (weak) homotopy equivalence) already in topological spaces, or in the the cateogry of spectra (which is an "extension" of $\mathbf{Top}$ to do homotopy theory).
As an analog, think of Mayer-Vietoris sequence. there is an underlying short exact sequence of the chain complexes, which is theoretically more important than the MV sequence itself, which is used for computations.
Another analog to chain complexes is that in both categories we have a notion of a mapping cylinder, which gives rise to long sequences of "suspensions" (this approach is encapsulated it the theory of triangulated categories and derived functors).