What is the meaning of the rule "all that dries is pure," and does it apply to clothes that are affected by something dry from an infant's urine or pre-seminal fluid, or to sofas that are affected by children's urine and then dry?

The statement mentioned is not a Sharia (definition) rule, unless it is intended to mean that a dry pure object that meets a dry impure object does not become impure. Impurity is not purified except by washing with pure water, and it is not purified by dryness except in specific cases. A nursing mother is excused from the impurity of the child after taking the necessary precautions, as is the view of the Maliki school. There are things that are purified without washing according to some scholars, such as the urine of a boy who has not been fed, for which sprinkling suffices.