You're visiting a friend's house and you've been there for a while. You say this because you think that your friend is probably ready for you to leave.
I should get going.
"Get ___ing" means to start doing something. It's used in casual speech. "I should get going" sounds softer and friendlier than "I should go".
"Get ___ing" is not a structure that can be used for just any verb. Only a few good examples spring to mind:
- get going (discussed above)
- get moving (often used when leaving your own house)
- get the ball rolling (to start a process or a project)
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