“Me” is the objective case, “I” is the subjective case. If the word is the subject of the sentence, use “I”: “Brian and I look purty in that there pitcher.” If the word is the object (just about anytime it’s not the subject), use “me”. “It [subject] is [verb] me [object].” Similarly, “That’s me.” or “That’s Brian and me in that there purty pitcher.”
The problem is we kinda lose track of what case we’re using when the sentence gets even a little complicated. After all, we’re not really consciously thinking about rules of grammar – we’re just talking and if it sounds right we go with it. So nobody gets too alarmed if we say “That’s Brian and I” – using a simple noun phrase rather than a pronoun caused us to lose track. Notice the difference if you said “That’s I and Brian.” That one is much more noticeable just because the pronoun comes first.
So although “That’s Brian and I” is technically incorrect, you hear that particular “error” a lot. And it’s what you get used to hearing that eventually determines correct usage. Someday there may even be a new “rule” that says “Use ‘me’ in the objective case unless the sentence contains the word ‘Brian’.”
“The departure of the church-going element had induced a more humanitarian atmosphere.”
Dorothy L. Sayers
Clouds of Witness