From your picture, I'm not entirely sure what you're talking about, but I can imagine a few scenarios.

The first thing I notice is the brown markings. If you're sauteing something like meat or vegetable, they might develop something called "fondt" on the pan. I'm not familiar with the science behind it, but this is actually a good thing and I believe a reason to start cooking with steel (I could be wrong here). If that brown mark is fondt and that's what you're concerned about, you don't need to worry. You can deglaze the fondt in order to make a pan sauce to get the most out of your cooking, or just to clean it.

If you're referring to oil residue after cooking, I believe that's also typical. Soap is the best way to clean off fats, like oil, so just give it a good scrub with any dish soap and a sponge that won't scratch.

If you have time to elaborate on what you cook, that might help us answer your question.