Actually, stainless steel does get stains that may be hard to remove. But 'stain' here refers to rust, which is something stainless steel rarely catches. Ever wondered why?
Why steel rusts
Steel is an alloy of iron and carbon. The carbon (which is less than 2% of the material) in the steel makes it harder than ordinary iron. But the iron in it is susceptible to rusting. When there's moisture in the air, iron atoms in steel react with oxygen to form iron oxides. These deposit on the surface as the red spots we call rust.
If you wash a rusted vessel in water, or scrub it with a scouring pad, some of the rust will come off. This leaves the iron below the surface now exposed to air. Again it will react to form more rust. Over time, the rust will go very deep into the vessel, and it may break.
If you use any steel items at home, it's important to keep them dry. Even if you wash them, you must wipe them dry; don't let them dry in the air.
Why stainless steel doesn't rust
Stainless steel is a different kind of alloy. Along with iron and carbon, it also has nickel and chromium in it. Nickel and chromium have an interesting difference in chemistry, compared to iron.
They too, react with oxygen to form their oxides. But there is a major difference. These oxides are very sticky, and they form a thin film around the stainless steel vessel. They do not dissolve in water, and do not come off even if you scrubbed the dish hard with a scouring pad.
Because of this, once a thin layer of oxide has covered the vessel, the rest of the material will be cut off from the air. It will therefore never rust, and remain as strong as ever.
Nickel and chrome are often alloyed with other metals. These are used to make things that are subject to heavy wear and tear, and also are at risk from oxidation in the air. Making them from nickel or chrome alloy prevents oxidation, and also adds extra hardness.
That's why coins which pass from hand to hand, are often made of nickel alloy. Car and truck wheels, which face a lot of dust, are often made from chrome alloys. Look around you, what objects can you see that are made of chrome or nickel alloys, including stainless steel?