The phrase, let that sink in, means “take time to think deeply about that.”
That, in the phrase, refers to some idea, fact, or piece of information. The word, in, means in your mind. To sink normally means “to go down, gradually, like a boat going down” but in this phrase, to sink in means to “to gradually be understood more deeply.” Depth in English is a metaphor for meaning; something that is deeper has more meaning.
So, let that sink in is a command, meaning “take time to allow yourself to understand this idea, fact, or information more deeply (with more meaning).”
This idiom can be used in many situations. People say it when they want listeners to understand deeply, instead of “letting it go in one ear and out the other”—the opposite of letting it sink in.
People may say let that sink in when they want you to care more about an issue or when they want you to think about the implications:
About half of Americans don’t believe in evolution; let that sink in!
America spends more than half of its budget on weapons; let that sink in.
This is a popular phrase, but be warned: it can sound pretentious—as if the speaker presumes to teach the listener.