That said with だって
When a sentence begins with だって, it refers back to the prior sentence, regardless of who said it.
だって isn’t a word itself, but rather a combination of だ and って.
The だ portion is the same as appears at the end of a noun sentence, a sentence which says “[something] is [noun]”.
An example sentence in English would be, “The animal living under the park bench is a cat.”
The って portion is a short form of the quote marker と and the verb 言う (meaning “to say”).
Adding the meaning of this って to our English sentence changes it to, “They say the animal living under the park bench is a cat.”
Any Japanese sentence that ends in だ can have って added to it to change it from “This is X” to “They say this is X.”
Note that “they” can be any other party, so it can be “He said” or “She said” and so on.
Usage
A speaker typically refers back to what they just said to provide context, their reason for their saying it. “I say that because…”
When said following another person’s line, it tends to be contrastive, such as “You say that, but…” However, it doesn’t have to contrast or negate the sentence before it. There’s nothing inherent in だって that requires a contrast. だって can be used to take what someone said and add information to it.
