The young police officer did all he could to put me at ease.

=The young police officer did all (关系代词that也省略)he could （do省略） to put me at ease.（动词不定式to put me at ease作目的状语）

=The young police officer did what (融合型关系代词）he could （do省略） to put me at ease.

年轻的警官做了他能做所有的事去让我放心/放松。

put sb at ease是某人放心/放松

Hearing his words put me more at ease.  

听了他的话,我心里清爽多了.