The grid creator makes very simple lines with vertices only on each end. You'll need to 'densify' the geometry (adding extra vertices) so that intermediate points can be projected and define the shape of the lines correctly.
For example, here's an unprojected map with a default grid from QGIS (5° intervals):
If we change the projection (Aitoff 54043), the lines still render straight but clearly the shape of Africa has changed:
Using the 'densify by count' tool from processing and adding 50 vertices per line, we get a new grid that seems to curve correctly:
Editing the vertices of one line of the densified geometry shows how many points define the shape along the way: