После разговора с учителем мне стало понятнее, как работает это правило.

Because после requires the genitive case.

• разговор = nominative singular
• разговора = genitive singular

So:

• после разговора = after the conversation

This is a very common pattern in Russian:

• после урока = after the lesson
• после работы = after work
• после встречи = after the meeting

So here, разговора is simply the form required by после.

Here с means with, and in that meaning it normally takes the instrumental case.

• учитель = teacher
• учителем = instrumental singular

So:

• разговор с учителем = a conversation with the teacher

Be careful: с can take different cases depending on meaning.

• с учителем = with the teacher → instrumental
• с горы = from the mountain → genitive

In this sentence, it clearly means with the teacher, so instrumental is used.

Because Russian often uses a dative experiencer in sentences about how something feels, seems, becomes clear, is easy/hard, and so on.

So:

• мне стало понятнее literally means something like to me it became clearer

This is different from a sentence with я as a normal subject, such as:

• я понял = I understood
• я лучше понял = I understood better

The version with мне is a bit less direct and more about your internal state of understanding changing.

Because this is an impersonal construction.

In Russian, expressions like:

• мне стало ясно
• мне стало легче
• мне стало интереснее
• мне стало понятнее

often use стало in the neuter singular past, even though there is no visible neuter noun. This is a standard impersonal pattern.

So стало here does not agree with мне.
It is just the normal past form used in this kind of structure.

Понятнее is the comparative form of понятный / понятно and means clearer, more understandable, or easier to understand.

In this sentence it works predicatively:

• понятно = clear / understandable
• понятнее = clearer / more understandable

So:

• мне стало понятно = it became clear to me
• мне стало понятнее = it became clearer to me

The comparative suggests that you understood it somewhat before, but now you understand it better.

Russian often leaves that unstated when it is obvious from context or explained by a following clause.

Here, the thing that became clearer is explained by the clause:

• как работает это правило

So the sentence structure is basically:

• After the conversation with the teacher, it became clearer to me [how this rule works].

Russian does not need a dummy subject like English it.

Yes, absolutely. That would also be natural.

Compare the nuance:

• Мне стало понятнее, как работает это правило.
  = It became clearer to me how the rule works.
  This focuses on the change in clarity.

• Я лучше понял, как работает это правило.
  = I understood better how the rule works.
  This focuses more directly on your act of understanding.

Both are good Russian. The original sentence sounds a bit more neutral and reflective.

Because Russian often uses работать not only for machines or jobs, but also for systems, methods, mechanisms, and rules in the sense of how they function.

So:

• как работает это правило = how this rule works / how this rule functions

This is very natural Russian. Similar examples:

• как работает система = how the system works
• как работает метод = how the method works
• как работает закон = how the law works

So even though English learners may first associate работать with to work in the sense of having a job, it is also widely used for functioning.

Because как работает это правило is a subordinate clause.

The main clause is:

• После разговора с учителем мне стало понятнее

And then Russian adds the clause explaining what became clearer:

• как работает это правило

Russian normally puts a comma before such subordinate clauses.

So the comma is required here.

Because inside the subordinate clause это правило is the subject of работает.

The clause is:

• как работает это правило

Literally:

• how this rule works

Since правило is doing the action of работает, it stays in the nominative case:

• это правило = nominative singular

If it had a different role in the clause, the case might change, but here it is simply the subject.

Yes. Russian word order is flexible.

You could also say:

• как это правило работает

Both are grammatically possible. The difference is mostly about emphasis and style.

• как работает это правило sounds very natural and neutral here
• как это правило работает can put a little more focus on это правило

In the full sentence, the original order is the most natural choice.

Yes. Russian often allows both a noun phrase and a full clause.

For example:

• После разговора с учителем мне стало понятнее, как работает это правило.
• После того как я поговорил с учителем, мне стало понятнее, как работает это правило.

The first version is more compact and slightly more formal or written.
The second version is more explicit: after I talked with the teacher.

Both are correct.