I was aware that workers were making demands for a pay raise.
I didn't know that my wife had demanded that my brother pay the money back.
This article will be posted during the busy end of the season, but I would appreciate it if you could read it.
This is like a fancy and humble way of saying to know. Sure, the kanji means suppose, but think of it this way... When you say "I suppose" it's like a humble way of saying "I know." You're not being as direct about it this way, and in Japanese, that makes you more humble and polite. So, that's why 存じる means to know.
The reading is the same as the one you learned with the kanji. If you remember the kanji reading, you'll have no problem remembering this vocabulary reading as well!