“How these words evolved” is a question about etymology.
The noun “silence” comes from Old French, which inherited it from Latin silentium. It is used in English since the Middle English period (the oldest references in the OED are from the early 13th century).
The adjective “silent” is a Renaissance borrowing directly from the Latin adjective silens (accusative form: silentem). It is attested in English since 1565. Silentem and silentium have a common root in Latin, but from an English standpoint they are two different words, borrowed at two different periods.
The verb “to silence” is a de-nominal derivation from the noun “ silence”. It is attested from 1616 onwards.