I just want to add that when using 'butt in' on a situation, we mean 'to become involved in a situation that does not concern you'. It is used in an informal context here, and 'interfere' is a synonym for 'butt in' in this case.
'I didn’t ask you to butt in on my private business.'
I also want to give you another expression (phrasal verb) which means 'interrupt a conversation'.
It is 'break in (on something). We use it to mean 'to interrupt' or 'disturb' something:
'She longed to break in on their conversation but didn’t want to appear rude.'
‘I didn’t do it!’ she broke in.