It's great that you make a Brocho, and say the declaration. But how does it allow you to cook for Shabbos on Yom Tov? What is it about it that makes it Muttar?
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On a Biblical level you'd be allowed to cook on a Yom Tov even for a weekday, so long as there is enough time left in the day that you'd actually be able to eat the dishes you're preparing, if you're so inclined.
The Eruv Tavshilin, then, is needed only because of the Rabbinical enactment forbidding preparations on Yom Tov for afterwards. It represents the beginning of your preparations for Shabbos; then, the rest of the cooking (or whatever other activity you do) is a continuation of that. This serves two purposes:
(Beitzah 15a and Rashi there; Shulchan Aruch Harav 527:1)
It's debated in the Gemara, but the simple answer as I recall it is basically, it's allowed to prepare from yomtov to Shabbos. But Chazal want to remind people that this is the special case for Shabbos, not that you can always prepare on yomtov for the next day. Making an Eiruv is the way we remind people of this.