Learn Hardcore Japanese: sensei no setumei ha wakariyasui to omoimasu. - I think the teacher’s explanation is easy to understand.

の is the genitive or possessive particle here. It connects two nouns—先生 (teacher) and 説明 (explanation)—to mean “the teacher’s explanation” or “the explanation given by the teacher.”

Why is は used after 説明 instead of が?

は marks 先生の説明 as the topic of the sentence (“As for the teacher’s explanation…”). If you used が (“先生の説明が分かりやすいと…”) you’d be focusing more on identifying or introducing that explanation as new information, whereas は sets it up as something already in the discourse that you’re describing or giving your opinion about.

What is the role of と after 分かりやすい?

This と is the quotative particle. When you express thoughts or indirect quotes with 思います (“I think”), you take the preceding clause in its plain form and add と to link it to 思います. So 分かりやすいと means “(that it) is easy to understand…” and attaches to 思います.

Why isn’t there a です after 分かりやすい?

In embedded clauses (those quoted by と), you use the plain (dictionary) form of verbs and adjectives, even in polite speech. Also, 分かりやすい is an い-adjective, so it doesn’t need a copula like だ/です in plain form. The politeness comes from 思います, which is in its polite form.

What kind of adjective is 分かりやすい, and what does it mean?

分かりやすい is an い-adjective meaning “easy to understand,” “clear,” or “straightforward.” It consists of the verb stem 分かり (“to understand”) plus やすい (“easy to do”), literally “easy to understand.”

How would you change the sentence to the past tense (“I thought the explanation was easy to understand”)?

You have two options:
1) Change only 思います to past:
先生の説明は分かりやすいと思いました。
(I thought “the explanation is easy to understand.”)
2) Also make the embedded clause past to emphasize the explanation itself was easy:
先生の説明は分かりやすかったと思いました。
(I thought “the explanation was easy to understand.”)

Why is the embedded clause in plain form even though the overall sentence is polite?

In Japanese, subordinate clauses linked by particles like と must use the plain (dictionary) form. Politeness is applied at the end of the main predicate—in this case 思います, which is in its polite form.

Who is doing the thinking, and why isn’t a subject like 私 stated?

The speaker (I) is doing the thinking. Japanese often omits obvious subjects—especially 私—when context makes them clear. Here, it’s understood that you’re saying “I” think that the teacher’s explanation is easy to understand.

Could you make a direct statement without と思います?

Yes. If you want to plainly state “The teacher’s explanation is easy to understand,” you’d say:
先生の説明は分かりやすいです。
Here you drop the quotative と and 思います, and use です after the い-adjective for politeness.