How am I suppose to browse through old contents on these boards?

Use this:

You don't. You're supposed to live in the now. That's the concept of the "lifestream" (not the FF one).

Why do you want to?

The normal answer here is to search for the content you want to see.

The search tool here isn't good enough for that yet, so probably the best answer is to go use google with a "site:boards.na.leagueoflegends.com" search.