What is the logic behind Tayammum for it really does not clean you?
In Islam, each directive has a form and a spirit. The form is its outward manifestation and the spirit its inner one. The form is dependent on external circumstances, whereas the spirit is not and only requires a person's sincere will. In ordinary circumstances, a directive is required to be carried out keeping in consideration both the form and the spirit. However, whenever it is not possible to carry out the directive in its required form (which as pointed out above is dependent on certain variables), Islam wants the directive to be carried out in some reduced or replaced form so that the spirit part can be kept alive. Consequently, if a person is not able to pray while standing he has been allowed to pray while sitting or lying down so that the spirit of the prayer is kept alive.
Tayammum when wudū is not possible should also be understood keeping in view this principle. It is meant to keep alive the spirit behind the directive of wudū. Of course it does not clean a person; however, it does make him pure in his heart if he knows its real philosophy.