First you’ll want to create a hex grid and have a polygon to match it to.
We’re going to make a selection of the polygons that do not overlap with the polygon, then delete them.
Start by going to Research Tools > Select by Location…
We’ll then select the grid layer as what we are selecting features from, the state polygon as the ‘compared to’ layer, and disjoint to get everything that doesn’t touch. We’ll set it to create a new selection (the default) and hit Run.
Look at that! Selected! We now want to delete these polygons. There are a few ways to do it, but one way: click the Pencil icon to turn on edit mode.
Hit the delete key. The selected polygons will disappear!
Then click the Pencil icon again to turn off edit mode. It will ask if you want to save your grid - say yes.
All set! Your shapefile now only has the hexagons that overlap with your shapefile.
Note: Your grid is probably a temporary layer, so you’ll want to right-click it and save it as an actual shapefile!