• 믿습니다 is in "declarative" mode.
To validate it, the English sentence would need a subject (you, he, she, or they). This is obviously not the case.
• Without a subject, the sentence given has to be either in the imperative or propositive mode.
Once you decide on the mode to be used, you then select the style (formal/casual) and the level of politeness (indifferent/respect).
• DLG uses: Imperative mode, formal + respectful style. Hence, 믿으십시오.
But any conjugation in imperative or propositive mode should be acceptable.
I have been learning Korean for the past 5/6 months.
DLG is my first and main source of learning at the moment.
Its lack of feedback especially on grammar questions has forced me to do a lot of search on the internet.
My favourite sites are:
• Morninglands (concise and easy to understand);
• Itaki (Good Youtube videos; great tips);
• Reddit & Quora (background knowledge)
• And of course, the Wiki pages (esp. for verb conjugation.)
Skip to point (3) if not interested in the nitty gritty of grammar.
• 믿습니다 -> declarative mode (used to deliver a statement)
나를(/저를) 믿습니다 -> You/he/they believe me (as you suggested)
• 믿읍시오 -> imperative mode (used to give instruction/order)
나를 믿읍시오 - Believe me!
(2). [Hon.] 믿으시다 [-(으)시다 suffix, used in reference to subject of the conversation who may not necessarily be the Listener]
• 믿으십니다 -> declarative [Honorific]
저를 믿으십니다 -> You/he/they believe me
• 믿으십시오 -> imperative [Honorific]
저를 믿으십시오 -> Believe me! (as seen in DLG example)
Note: With honorifics, the subject of conversation is someone you, the Speaker, hold in very great respect whether by rank or by age. It is therefore a must to use "the humble 저".
(3). In short, your mistakes are in the verb mode (mood) and the level of respect (Honorific) used.
나를(/저를) 믿습니다 is in declarative mode, non honorific.
저를 믿으십시오 is in imperative mode, honorific
믿다 (base form) => 믿 + (-으)시다 (added honorific) =>
믿으시다 (base form with honorific).
And in imperative mode:
믿으십시오 is high formality (formal, polite), plus honorific.
Less formality (with honorific) would be
믿으시라 (formal, non polite: impersonal)
믿으셔 (casual, non polite: intimate)
믿으세요 (casual, polite: familiar )