Learn Hardcore Japanese: bunsyou wo mizikaku kaku houhou wo sensei ni kikimasita. - I asked the teacher for a method to write sentences concisely.

The first を marks 文章 as the direct object of 書く inside the relative clause 文章を短く書く (“to write sentences more concisely”).
The second を marks 方法 as the direct object of 聞きました (“asked”). In other words, you asked about the method (方法).

How does 文章を短く書く modify 方法?

In Japanese, a verb phrase can directly precede and modify a noun, just like an adjective. Here 文章を短く書く (“to write sentences concisely”) is a relative clause that describes 方法 (“method”). Together they form “the method of writing sentences concisely.”

Why is 短く used instead of 短い?

短い is an i-adjective meaning “short.” To turn it into an adverb that modifies the verb 書く, you change 短い to its adverbial form 短く. So 短く書く means “write (something) in a short/concise way.”

What does the particle に do in 先生に聞きました?

With the verb 聞く meaning “to ask,” に marks the person you ask. So 先生に聞きました means “I asked the teacher.”

Could I say 先生から聞きました instead of 先生に聞きました?

Not if you mean “ask.” 先生に聞く is correct for “ask the teacher.” 先生から聞く would mean “hear from the teacher” or “learn from the teacher,” emphasizing receiving information, not actively asking.

Why is the main verb 聞きました placed at the very end?

Japanese generally follows Subject–Object–Verb (SOV) order. Modifying clauses (like 文章を短く書く方法) and objects (方法) come before the verb. That’s why 聞きました ends the sentence.

What exactly does 方法 mean and how is it used here?

方法 means “method” or “way.” In this sentence, it’s the thing you “ask about.” You’re asking your teacher for the method of making your writing shorter.

How would I form a similar sentence to ask about another skill, e.g. “how to pronounce words correctly”?

Use the same pattern:
[Noun + を + Verb + 方法] + を + [Person] + に + 聞く.
For your example:
単語を正しく発音する方法を先生に聞きました。
(“I asked the teacher how to pronounce words correctly.”)