1 At that moment I heard the sounds of the young people I'll turning to the cottage.

at that moment为介词短语作时间状语。

2 We like them in the same way that we like pretty curtain material.

in the same way为介词短语作方式状语修饰like. that从句为定语从句修饰名词词组the same way. that为关系副词(指代先行词the samen way), 在从句中作状语。

3 I want to see him the moment he arrives .

the moment虽然形式上是一个名词词组，但已经被重新分析为引导时间状语从句的连词，大约等于as soon as. the moment he arrives为时间状语从句，修饰不定式to see. 即to see him the moment he arrives合起来是一个不定式短语，作want的宾语。