seonsaengnimi dasi seolmyeonghae jusigo naseoya i munjereul ihaehaesseoyo.

A natural breakdown is:

• 선생님이 = the teacher
  • subject marker 이
• 다시 = again
• 설명해 주시고 나서야 = after explaining it for me/us — and only then
• 이 문제를 = this problem/question
  • object marker 를
• 이해했어요 = understood (past, polite)

So the structure is basically:

[After the teacher explained it again] + [I understood this problem].

Notice that Korean often leaves out things that are obvious from context, especially I/me and the thing being explained.

-고 나서야 means only after doing X, or not until after doing X.

So:

• 설명해 주시고 나서야 = only after (the teacher) explained it for me/us

The important nuance is that the understanding happened only then, not before.

Compare:

• 설명해 주시고 나서 이해했어요 = I understood after the teacher explained it
• 설명해 주시고 나서야 이해했어요 = It was only after the teacher explained it that I understood

The 야 adds emphasis and often suggests that before that point, the result did not happen.

The 야 adds a feeling of only then, not until then, or finally after that.

Without 야, the sentence would just say the second action happened after the first one.
With 야, it implies something like:

• I tried before, but it didn’t work.
• That explanation was the key.
• Understanding happened only at that point.

So -고 나서야 is stronger and more pointed than just -고 나서.

설명해 주시고 comes from 설명해 주시다, which means to explain for someone, to kindly explain to someone.

There are two ideas here:

• 설명하다 = to explain
• -아/어 주다 = do something for someone / as a favor or benefit

So 설명해 주시다 does not just mean explain. It means something closer to:

• kindly explain it to me/us
• explain it for my/our benefit

If you said 설명하시고, it would simply mean explained, without that benefactive nuance.

The -시- is the honorific marker. It shows respect toward the teacher.

Here’s what is happening:

• 주다 = to give / to do for someone
• 주시다 = honorific form of 주다

Because the subject is 선생님 and teachers are typically spoken of respectfully in Korean, 주시고 is very natural.

So 설명해 주시고 includes both:

• the idea of doing something for someone
• respect toward the teacher

이/가 marks the subject, while 은/는 marks the topic and can add contrast or special emphasis.

Here, 선생님이 is a neutral way to say the teacher is the one who did the explaining.

• 선생님이 다시 설명해 주시고... = The teacher explained it again...

If you used 선생님은, it could sound more like:

• As for the teacher, they explained it again...
• or it could set up contrast with someone else

So 이 is very natural here because the teacher is simply the subject of that first clause.

The subject of 이해했어요 is not stated, but it is understood as I.

So the sentence is really:

• 선생님이 다시 설명해 주시고 나서야 [제가] 이 문제를 이해했어요.

In Korean, subjects are often omitted when they are obvious from context.
Here, the teacher is the subject of the first action (explained), but the speaker is the subject of the second action (understood).

That kind of subject shift is very common in Korean.

Korean often omits objects when they are clear from context.

After 설명해 주시고, the missing object could be something like:

• 이 문제를 = this problem
• 그 내용을 = that content
• 이걸 = this

Because the next clause says 이 문제를 이해했어요, we naturally understand that the teacher explained this problem or something related to it.

So the sentence does not need to say the object twice.

다시 means again.

It modifies the teacher’s action:

• 다시 설명해 주시고 = explained again

This suggests there was probably an earlier explanation, but it was not enough for the speaker to understand. Then, after the teacher explained it again, the speaker finally understood.

So 다시 fits very naturally with the nuance of -고 나서야.

를 is the object marker. It marks 이 문제 as the thing that was understood.

• 이 문제를 이해했어요 = understood this problem

The verb 이해하다 takes an object, so 를 is the normal choice here.

Also, 문제 can mean:

• a math/problem-solving problem
• a test/exam question
• an issue or matter

The exact meaning depends on context, but grammatically it is the object of 이해했어요.

이해했어요 is:

• past tense
• polite speech (해요체)

It comes from 이해하다:

• dictionary form: 이해하다
• past polite: 이해했어요

This is a very common everyday polite style. It is respectful enough for normal conversation, class situations, and most non-intimate settings.

Yes, very strongly.

Because of -고 나서야, the sentence strongly suggests:

• before that explanation, I didn’t understand
• the repeated explanation was what finally made understanding possible

So the nuance is not just a sequence of events. It is more like:

• I only understood after the teacher explained it again
• It wasn’t until then that I understood

That implication is one of the most important things this grammar adds.