How does exploring soil, purified sand, or dust of Tayammum make a person cleaner?

There is something, in my humble opinion, I really don't understand regarding Tayammum. According to the law, it's possible for one to **clean** himself by exploring soil, purified sand, or dust? (in case he doesn't have water). Let's assume that one has already clean hands and face and now he has to put them into the sand, how does it make them cleaner? I'm a bit confused by that.