Four factors affect depth of field (the part of an image that is 'acceptably sharp')
- circle of confusion
- iris diahragm
- focal length
- subject distance
To simplify a bit, depth of field tables/calculators use a standard value for circle of confusion (CoC) for each sensor size. DoF is directly proportional to CoC. FF sensors have a larger CoC, therefore they have a deeper DoF when you hold everything else constant. The common and useful proposition that larger sensors have shallower DoF applies to a shot framed identically. To do that with the same lens and f-number on a smaller sensor, you'd need to move further from the subject. That 'outweighs' the effect of CoC, and so when you take the same picture (i.e. same subject framing) with the same lens/aperture on a FF camera vs. crop camera, the DoF is shallower in FF.