saying "etefaghan" is like saying "as a matter of fact", "as it happens" [though not the exact translation]. It's for emphasis as our friend mentioned in the comment.
in this sentence it's like saying "as a matter of fact I am invited too"
"manam" is spoken form, of "man (I)+ ham(too)"
you can consider these contexts:
A: I'm not going to the party tomorrow?
B: Why? is it because she didn't invite you?
A: Etefaghan manam davat shodam. (as a matter of fact I am invited too), but I have a big test I need to study for.
A: Sara invited me to go to their family farm tomorrow. I'm looking forward to it.
B: Etefaghan manam davat shodam. I guess I'll see you there then.