Tearing the fabric does not straighten the fabric...it gives you a crosswise thread at each end of the piece. Then, you have to stretch the fabric to line up those ends and the selvage edges in order to straighten the fabric. In making garments, cutting a piece off-grain will end up with a piece that doesn't hang straight. Or, if a curtain is cut off-grain, it will hang crooked.
Tearing works for most plain-weave fabrics. When the fabric won't tear, you pull a thread and cut along it to find the grain. Grain refers to the length-wise (warp) and cross-wise (woof) threads. When they get off-grain on the bolt and you cut across, you are cutting across several threads...not along the weave of one thread.