My parents used to have an almost identical meat grinder that we used not all that frequently back in the 80's and 90's. It was old then. The reason I'm telling you this is because it was often very rusty and had to be scrubbed clean before use. You can see some rust still in the threads of the piece in the bottom right of your pic. This is because the chrome has very slightly chipped off.
However, the blades and the perforated plate are bare metal and will rust quickly, unless properly treated. In this case, proper treatment should have been a light coating of a vegetable oil. Unfortunately, vegetable oils can decompose after a while, so either that happened or the previous owner used an oil that wouldn't decompose, like kerosene, WD-40, or another light machine oil. This can happen if it's actually a brand new piece that's been manufactured, rather than a "new to you" purchase.
The way to fix this is to use a soap that's very good at removing grease (such as dish washing soap). Since this is cast iron, it has all kinds of nooks and crannies for the oil to hide. Be sure to use plenty of soap, warm to hot water, and let it soak for a couple minutes before you scrub it really well with a bristle brush. A scratch pad, sponge, or a wash cloth aren't going to get into everywhere as well as the bristles. The link below also suggests using white vinegar to help remove the smell. The vinegar should also help break down any oils leftover.
FYI, when I finish working on my cars engine and my hands are covered in grease and are completely black with nastiness, I use WD-40 to remove most of it. After that, I use a pumice soap and a fingernail brush to help remove the rest of the crud as well as the WD-40, and then I use another, de-greasing soap to finish removing the WD-40. I normally use Dawn dish soap in all my liquid hand washing dispensers because of this. At this point, my hands are generally as clean as they were before I started the repair.
Because of my method of cleaning my hands, I know that it should work quite well for the metal blades.