ように頼む / 言う

The phrase ように頼む / 言う is used to indirectly quote a request or command that was made. It's likely evident from the verbs themselves, but ように頼む translates more naturally as "to ask someone to [verb]" while ように言う is more like "to tell someone to [verb]."

The structure used here is generally [present tense verb] + ように頼む / 言う. An affirmative ように頼む request is often preceded by a verb in the てくれる form. Negative requests or commands simply use the verb in the ない form.

先生は学生に「漢字を覚えなさい」と言いました。
The teacher told the students, "Memorize the kanji."

先生は学生に漢字を覚えるように言いました。
The teacher told the students to memorize the kanji.

These examples should provide a better understanding of what this phrase actually does. Rather than directly quoting what the teacher said, we're giving a secondhand account (essentially paraphrasing) of what he told them.

パーティーをしていたら、隣のアパートの人に静かにするように言われてしまいました。
When we had the party, we were told by the people in the next apartment to be quiet.

アヤトに買い物に行くならきれいなドレスを買ってくれるように頼んだ。
I asked Ayato to buy me a beautiful dress if he goes shopping.

ウェイターにデザートメニューを持って来てくれるように頼もうか？
How about we ask the waiter to bring over the dessert menu?

スバルは学校の後に家に帰るように言われてた。
I was told by Subaru to come home after school.