A collection of letters, diaries, and the like is not, technically speaking, an archives. It is, rather, a manuscript collection. An archivist is the person who decides what to save and what to throw away. Archivists also answer questions and provide for the physical preservation of documents. If you believe that what happened to you yesterday, or last week, or last year, is important then you understand why an archives is important.
Archives preserve the past.
By preserving evidence. A rational being is one who requires evidence to believe or understand. Archives preserve evidence so that we, as a society, may understand what has happened in the past. Without evidence, no one would know what to believe about yesterday, last year, or even a thousand years ago. Society, too, has a memory and an archives is where society's memories are kept.