This follows from the following claims. Let me know if the last two seem to need further fleshing out. For me this stuff requires visualizing the zeroes in suspension squares as upward- or downward-pointing cones on the object being suspended.
- When you restrict $j_!i_*(f)$ along $d^1\times d^2,$ you get $(0,0)_*X.$
- When you restrict $j_!i_*(f)$ along $d^0\times d^1,$ you get $\sigma^*(0,0)_*Y,$ where $\sigma$ is the nontrivial automorphism of $\ulcorner.$
- By definition of the cogroup structure on suspended objects, a map $(0,0)_*X\to \sigma^*(0,0)_*Y$ evaluating to $f$ under $(0,0)^*$ induces $-\Sigma f$ when extended to the suspensions.
- Since the natural transformation $\alpha:d^1\times d^2\Rightarrow d^0\times d^1$ (pretty sure Groth has the variance backwards here) has $(0,0)\to (1,0)$ as its component at $(0,0),$ the component of $\alpha^*$ at $j_!i_*(f)$ evaluates to $f$ under $(0,0)^*.$