。
Don't cut into our conversation.
Literal
We [of-の] talk [in-に] don't-cut-in [emphasis-よ].
割り込む ('to cut in, to intrude') takes に for the target of intrusion. A compound verb: 割る ('to split') + 込む ('to go into'). Vivid imagery — literally 'splitting one's way into.' Similar meaning to 邪魔をする but with the specific nuance of insertion rather than general interference.