How to understand get?
If you have ever seen the Star Trek movies where the captain of the starship says "make it so" then you are well on your way to understanding what "get" means in phrases like "get on a list" or "get on a train".
To "get on a list" means "to cause your name to be on the list". You "make it so" by taking some action. You might write your name down yourself, or send an email asking someone to put your name on the list. But you take some action to "make it so".
To "get on a train" means "to take action to put yourself on a train".
The action could be as simple and easy as walking through the open train door, or it could involve a complicated plan:
I don't have a ticket but I must get on that train!
Whatever the action happens to be, you "make it so" that you are on the train.
Thus a phrase like "I must get going", which means "I must be on my way, that is, I need to leave (this place)", the "raw" meaning is "I must make it so I am going", that is, I must not just be thinking it is time to leave, I must actually begin to leave.
P.S. Since OP has added a comment to say that the phrase comes from a song lyric, "Mama, they say I'm a terrorist, what? I did nothing wrong, but I got on a blacklist" we need to address another use of get/got.
Why are you making cupcakes for the kindergarten class? You don't like to bake.
-- I don't know. Somehow I got on the list.
This usage isn't as different as it may seem. The person's name didn't put itself on the list. And yet the fact of the matter is that the name is now on the list, and it was effected or "made so" by an unmentioned mysterious agent, or agency, in the case of the song lyric.