Nikon does this thing where when you mount a DX lens it (automatically if the Auto DX Crop option is enabled) uses a reduced portion of the sensor — the "DX mode" mentioned in the question. This addresses the image-circle issues raised in the other answers here.

Both Sony and Nikon offer this sort of compatibility mode, but because of physical limitations, you can't even mount the crop-factor Canon lenses on full-frame cameras. (And Pentax doesn't have a full-frame offering.)

Anyway, the lens is a 35mm lens and it always stays a 35mm lens regardless of sensor format, but the field of view on a DX (APS-C) camera is the same as the field of view through a 52.5mm (1.5×) lens on a FX camera. So good so far.

When you use DX mode on your FX camera, the edges of the sensor are ignored, just as if they were cropped away. (In a way, this is exactly the same as it is on a DX-format camera, except on those it's permanently away.) Since the sensor size is now for all intents and purposes the same as in a DX camera, this gives the same narrower field of view, again equivalent to an FX-format 52.5mm lens across the whole FX sensor.

If you use an FX-format 35mm lens to take a picture, and then crop out the middle 2/3rds (that is, 1/1.5 — the crop factor!), the field of view of the resulting crop is that same 52.5mm. Or, if you force DX mode on with that lens mounted, same thing. No magic is happening — you're just discarding the edges of the field of view.