The doctors were encouraged that, when he spoke about his mother, his expressionless mask slipped slightly, and he looked as if he was struggling not to cry. (original)
Question : Is the original grammatically correct?
Would you tell me how to understand the word "that" (conjunction) here?
I guess :
The doctors were encouraged that, when he spoke about his mother, his…
=
Someone encouraged the doctors saying that or by saying that, when he spoke about his mother, his…
Is my guess grammatically correct?

Question

The doctors were encouraged that, when he spoke about his mother, his expressionless mask slipped slightly, and he looked as if he was struggling not to cry. (original)
Question : Is the original grammatically correct?
Would you tell me how to understand the word "that"  (conjunction) here?
I guess : 
The doctors were encouraged that, when he spoke about his mother, his…
=
Someone encouraged the doctors saying that or by saying that, when he spoke about his mother, his…
Is my guess grammatically correct?

Answer

Wow, this is a hard one to analyze! The sentence confuses me because it provides no reason why the doctors were encouraged. Instead, it describes a situation that caused the doctors to become encouraged without stating why this situation would have been encouraging.

The sentence should have been written like this:
"The doctors became encouraged when he spoke about his mother, his expressionless mask slipped slightly, and he looked as if he was struggling not to cry."