Q:

What is meant/implied by "じゃないんだからね"?

I noticed today, while playing Grisaia no Kajitsu (English version, since my Japanese is really bad), that when I click on a menu option while using Michiru's voice to voice the menus, she ends her sentence with "じゃないんだからね" (one example being clicking "Extras," which causes Michiru to say "おまけじゃないんだからね"). I'm not entirely certain that she uses the ん sound, but it sounds like it to me. It sounds as if she is telling me that there is no omake, but I feel like I'm missing the importance of the "んだからね". She is supposed to be a tsundere (well, a fake tsundere anyway), so that likely plays into this. 
Here is a short clip of her saying it (4 seconds, made with Windows 7's sound recorder), so that you all may tell me that I terribly misheard everything and should give up on Japanese forever.
I converted it to a wav (using online-convert.com), so it should play without being downloaded, I think.

A:

You heard it quite right; she is saying 「おまけじゃないんだからね！」. 
Where you would normally hear something like this from a ツンデレ is in a scene where she does something for a guy and then follows it up with 「べっ別に好きってわけじゃないんだからね！」 "I-It's not like I like you or anything!"
In my opinion, it doesn't really make that much sense in this context, but saying 「〜じゃないんだから」 excessively is basically part of the ツンデレ trope, and I guess they are just extending it to somewhere where it sounds completely ridiculous for fun. She is basically saying "It's not like this is an extras screen or anything!" or "It's not like these are extras or anything!!" or however you might want to interpret it.

A:

so that you all may tell me that I terribly misheard everything and should give up on Japanese forever

You heard it all exactly correctly including the ん and should not give up on Japanese :)

It sounds as if she is telling me that there is no omake

Yep. Or, rather, that these are not omakes (extras). ("There is no" would be がない or はない rather than じゃない).
It's a standard tsundere trope/behavior to stubbornly deny (out of embarassment and a desire to maintain the "tsun" part) that a situation is a certain way, especially when doing favors or giving compliments. Like 「あんたの為じゃないんだからね」 ("I'm not doing it for you, you know[, so...]!").
The だから ("because/so") omits a second half -- "...so don't go thinking that I am / don't thank me / etc". The addition of ね actually makes it even more "tsundere" because ね seeks agreement, to a degree, somewhat like "okay?!". It makes the statement lose a little bit of confidence (did the bluff work?) which increases the effect.
Michiru's thing is that she hasn't got the whole "pretending to be a tsundere" thing quite figured out. She's not just denying things like a normal tsundere, she's overacting and overusing this mechanism anywhere and everywhere, denying even things that are blatantly obvious and which don't hurt her at all in the first place. 
Enjoy Grisaia :)

