The second character is using the word "really" as a verb, meaning "to say 'really' to somebody in order to express disapproval". Thus the phrase "don't 'really' me in that tone" means "don't say 'really' to me in that tone".
Saying "really" as the first character has done here is almost certainly done in a disapproving manner, which the second character is finding to be rude or impolite.
The second character then also complains about the use of the word "ungrateful" to describe them.
I think the meaning here is that the first character thinks the second character is ungrateful and has emphasised that by saying "Really" in a disapproving tone, something which is quite common in certain forms of English. The second character objects to this, but refers to it in a very concise way by their use of "really" as a verb.