I thought "spare me with ..." means "don't bother me with the details of ...". Does it?

But according to my friend I am wrong. What do you think?

"Spare me" can be used to refer to anything that isn't wanted.

As in, "spare me your poppycock", "spare me your pedantic ways", "spare me your grandiloquence" etc.

"Spare me" just means "let me not hear/see/experience something".

You are correct. However, I think this is an instance of an unnecessary preposition. I would say, "Spare me the details".

If your friend doesn't believe you, tell him or her to read these lyrics¹ by the punk band The Offspring:

Warning: racy lyrics

In an absolute sense, your friend is correct: "spare me with..." doesn't really mean anything. Thus it doesn't mean "don't bother me with the details of...", but this is about as useful as saying that it doesn't mean "the sky is blue".

What you were doubtless intending to say is "spare me the details [of...]", which does indeed mean "don't bother me with the details [of...]". "Spare me [x]" is an idiom meaning "I don't want to be bothered with [x], so kindly leave [x] out of any further conversation."

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