Imagine you have an FX camera (or old film camera) with a 50mm lens, and take a picture. Then in post processing, you crop out the edges. You would still have an image with the same perspective of the 50mm lens, but by cropping you've effectively zoomed in on the subject/middle of your image.
This is basically what happens with a cropped sensor. Same 50mm lens, same image projected onto the sensor, but the sensor being smaller (DX), you only get a smaller part of that projection.
So the perspective and DOF is like a 50mm lens, but the subject area you get is roughly like having a 75mm lens (on a film or FX body) due to the zooming/cropping effect.
As an example, if you are shooting a full body shot of a person, or a statue in a museum, film camera and 50mm lens, and then switch to a DX body and 50mm lens, you will have to stand further back in order to get the same framing. If you want to be the same distance as you were with your FX + 50mm lens, you'd need something like a 35mm lens on a DX body.