See https://math.stackexchange.com/questions/415787/motivation-of-stable-homotopy-theory
Fiber and Cofiber sequences Agree There is a theorem of Blakers and Massey, sometimes called homotopy excision which says: even though taking homotopy groups does not take cofiber sequences to long exact sequences, it does do so in a certain range depending on the connectivity of the cofiber sequence.
In particular, if we keep suspending the cofiber sequences (in which case we’re asking about stable homotopy groups) then this is the case. This turns out to be enormously useful (think about every time you make an argument in homology using the long exact sequence of a pair… you can now make that argument with stable homotopy groups, or any other nice spectrum, etc.)