For example: "Instagram stories got me like, you all need to listen to better music".
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It means 'Makes me think and/or do'. Such as in the example, this person is reading through Instagram and posts and has come to the conclusion that the music they listen to is bad, and thinks (or has him like) they should listen to better music.
For example, if someone was to say "Your outfit got me like someone punched me in the gut" as a way to say someone's outfit was ridiculous. The phrase is also widely used to compare your feeling to another similar feeling, but something greatly exaggerated. Firstly, the 'like someone punched me in the gut' is comparing how the person stating the sentence is laughing so hard they cannot breathe, similar to being punched in the gut. the 'got me like' is used as a comparison to another exaggerated situation.