When your elderly relative needs to enter some sort of long-term care facility—a moment few parents or children approach without fear—what you would like is to have everything made clear.

本句中的二个破折号之间是一个名词词组，为when从句的同位语。如果在前面加上which is 就是一个非限制性定语从句，其先行词为when从句的内容。

非限制性同位语通常由逗号，冒号或破折号引出。本句的二个破折号也可以改为二个逗号：

When your elderly relative needs to enter some sort of long-term care facility, a moment few parents or children approach without fear, what you would like is to have everything made clear.

a moment （that） few parents or children approach without fear

a moment后面为省略了关系代词that的定语从句，that为approach的宾语。

to have everything made clear为have something done结构，意为把一切都弄清楚（即让养老结构把一切都解释清楚）。