3. Concocting A Plan
It's argued that here, off-camera (and off-page, I guess, if we're including the books), the two came to an agreement; the eagles would usher Frodo to Mordor as it's the safest option, but nobody can know except them. It's not a perfect plan, the Ringwraiths and their Fellbeasts wouldn't let them just waltz over without a fight, but with increased numbers and the element of surprise, it would be far safer (and quicker) than going on foot.
It was essential that nobody else could know about what the actual approach was going to be though, as if Sauron caught wind of the air-assault, Gandalf and his companions wouldn't stand a chance.
This is why he mentions nary a peep of it to anyone, even Frodo, as anybody with the knowledge of this Hail Mary approach could ruin the necessary element of surprise, voluntarily or not. Gandalf himself knows the burden of knowledge more than any other as well, as he was the one forced to reveal the location of the ring to Sauron's armies.