1The humble (kenjougo) form of "to know," used when speaking modestly to a superior or business contact.
お噂はかねがね存じております。
I have long known of your reputation.
詳細については存じませんので、担当者に確認いたします。
I don't know the details, so I will check with the person in charge.
その規則は存じておりますが、念のため確認させてください。
I am aware of that rule, but please let me confirm just to be sure.
2The humble form of "to think" or "to believe," used to modestly state one's own opinion.
私もその意見が正しいと存じます。
I too believe that opinion is correct.
来週までにご返答いただければと存じます。
I would appreciate it if you could reply by next week.
大変光栄に存じます。
I feel it a great honor.