What are the best ways to understand core issues your boss is facing when you are a new employee and you're not directly exposed to that knowledge?

if you don't know ,you will not tell him i don't know just you will learn how we do it and don't tell anybody i'm still learning.

you have to take care for the learning work issues then you should to make yourself hard worker especially in the beginning. 

Every job title has a job description. You can search his job title and description and figure out from there. Some subordinates will not tell you and leave you in the dark to figure it out. Research your boss's job description and put2 and2 together.

Awareness or understanding on some one /some thing is must. Boss always right is proverb .. Boss is allowed to tell your opinion / suggestions  then to tell or create the such atmosphere..  Don't trying as your own. Majority bosses are currently is friendly manner. so, you have to tell your opinion with out hesitation and make sure that the problems knowing yourself,.  or boss is asking you, you are not familiar  then tell politely that' sorry' '/no idea'. you are not trying  your own ,it will leads  unnecessary problem some time. 

This is my suggestion. Thanks.

Try to find the issues from other employees and if they don't know then go to your boss and ask him/her personally.