In the original book of Conley Index Theory: Isolated Invariant Sets and the Morse Index chp3.3, p6, Charles Conley mentioned that Morse Index of an isolated invariant set is defined as the homotopy equivalence of pointed spaces which is created by collapsing exit point set of isolated neighborhood to a point.
For a source critical point, its isolated neighborhood is an interval containing it, with two endpoints as the exit point set. So pointed 1-sphere is a representative of its Morse Index.
However, for a sink critical point, The exit point set is $\emptyset$. In the book ,$\emptyset$ is collapsed to a point to get a 2-points space, i.e. 0-sphere. How that happens?
It is just a definition that makes everything smoother. See the discussion here: https://mathoverflow.net/questions/118574/conley-index-for-isolated-invariant-sets-with-no-exit-points .
An index pair of a one dim sink is a pair of an interval (e.g. [-1,1]) and the empty set (no points are flowing outward). Then the Conley index is $([-1,1]/\emptyset,[\emptyset])$ which is $([-1,1]\cup \{*\},\{*\})$ as a pointed space. Note that this is homotopy equivalent to a space with two points, which is the $0$-sphere.