What did they mean when they said "Take her and go, as Hashem spoke"?
Seforno: they meant that Eliezer could actually take Rivkah and leave, even without their permission, as Hashem had decreed before Yitzchak was born! 1
Ha'Emek Davar: What Hashgachah shows, it is as if Hashem said it.
Why did they need a special reason to agree? Without the proof, it was good for Rivkah!
Ha'Emek Davar #1: It is difficult to go to another country (Kesuvos 110b).
Ha'Emek Davar #2: To marry a paternal relative is not a sign that this is the destined match, so they needed another proof.