Learn Hardcore Korean: sikdangeseo chingureul mannayo. - I meet a friend at the restaurant.

에서 is the locative particle that marks the place where an action occurs (“at the restaurant”).
에 marks a static location or direction (“to”/“in”), but when you’re describing an action taking place somewhere, you need 에서.

What role does 를 play in 친구를? Why is it necessary?

를 is the object particle. It marks 친구 (friend) as the direct object of the verb 만나요. Without 를, you wouldn’t know that “friend” is what’s being met.

What does 만나요 mean, and how is this form created?

만나요 is the present-tense polite form of the verb 만나다 (to meet).
To form the polite style, you take the verb stem 만나 + -아요 = 만나요.

Why is there no subject in the sentence? Who is implied to be doing the meeting?

Korean often omits the subject when it’s clear from context. Here, the speaker (I) is implied, so the full idea is “I meet my friend at the restaurant.”

Why does the verb come at the end of the sentence?

Korean follows a Subject–Object–Verb (SOV) order. That means the verb always comes last, after the subject and the object.

How can we change this sentence to a more formal level of speech?

Replace the polite ending -아요/-어요 with the formal polite -ㅂ니다/-습니다.
만나요 → 만납니다, giving:
식당에서 친구를 만납니다.