After Boromir is blinded by the greed of Men and tries attacking Frodo for the Ring, Aragorn finds Frodo and tells him to keep the Ring, saying he'll follow Frodo into the fires of Mount Doom.
This scene doesn't happen in the book; it's an invention of Jackson & co.
To be more precise: the scene with Boromir does happen, but Frodo gets clean away without meeting Aragorn (or anyone else).
There is no "sudden deep care" in the book. The companions are for the most part chosen (rather than volunteering, as in the movies) and they actually have no compulsion to follow Frodo to Mount Doom. In fact it says of Legolas and Gimli:
They are willing to go at least to the passes of the Mountains, and maybe beyond.
But on the other hand, of Aragorn and Boromir:
'I would have begged you to come,' said Frodo, 'only I thought you were going to Minas Tirith with Boromir.'
'I am,' said Aragorn. 'And the Sword-that-was-Broken shall be reforged ere I set out to war. But your road and our road lie together for many hundreds of miles. Therefore Boromir will also be in the Company. He is a valiant man.'
Therefore of those who set out on the quest, we have the following breakdown:
- Aragorn: going to Minas Tirith.
- Boromir: going to Minas Tirith.
- Legolas: going at least as far as the passes of the mountains.
- Gimli: going at least as far as the passes of the mountains.
- Frodo: going to Mount Doom.
- Sam: (presumably) going to Mount Doom.
- Merry: unknown.
- Pippin: unknown.
- Gandalf: unknown.
Aside from all of that, the company were journeying together for some time, endured dangers together, and so it's only natural that a friendship built up between them. We see the same in the friendship between Legolas and Gimli.