What does it mean when someone says "indulge me"?

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It is a way of asking for permission to do something, usually talking. For example, person A wants to tell a story, but person B isn't interested. Person A might say, "Please just indulge me for a couple minutes." Or a teacher might say to her class, "Please indulge me if I don't follow our usual lesson plan today." In other words, the teacher is asking her class to be patient with her and not complain.

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Go ahead and tell me, I’m interested

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Indulge me also refers to sarcasm. "humour me"

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If they say it, just that, "Indulge me," it's in answer to an objection.

"But it will take too much time!"
"But you already know this!"
"But we did it ten times already!"

It acknowledges that the request might be going overboard but please do it anyway.

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You could also say "gratify/humor me"