To some extent you could suggest that Peter Jackson was testing his Aragorn character and proving that he was strong willed enough to be the King.

Frodo sees Boromir struggle to resist the Ring and later challenges Aragorn asking:

“Can you protect me from yourself? Would you destroy it?”

Peter Jackson's inclusion of this made up scene seems to challenge Aragorn's purity and strength and allows him to have a significant role in the separation of Frodo (and Sam) and the Three Hunters. It also lessens Frodo's role (as is done throughout the film) as more of a passive character to other people's ideas/motivations. Aragorn giving up the Ring and letting Frodo go demonstrates his ability to realise that this is not his quest, and that Frodo needs to continue on alone. Also that he wouldn't have the strength to resist it and to keep the others in the party from having to face the challenge.

This seems to be supported by the Director's Audio commentary from the Fellowship of the Ring (between writer Philippa "Pippa" Boyens and director Peter Jackson)

Philippa: No, that was Fran and I. Remember one time when we were just like, “What the hell is all of this? What the hell is all of this?” and we realised that… One of the reasons for this particular scene is that we felt very strongly that early on – especially in earlier drafts, before we started filming – that these two great characters – who go on to carry the main story threads for the rest of the films – needed this moment together. And actually –.

Peter: [interrupting] And also it juxtaposes exactly what’s happened with Boromir in the sense that [Philippa: Exactly.] there’s one Man who was tempted by the Ring [Philippa: Mmm.] and –

Philippa: Nobody’s…

Peter: – couldn’t resist and here is another Man who is tempted by the Ring at this moment, and he does resist it. He is… He has got the strength to push it away, so it’s also important for Aragorn because, in a way, this actually proves something to Aragorn himself: that Aragorn can see that he does have the power to reject the Ring when it’s offered to him.

Philippa: Yeah.

Peter: And that leads Aragorn, then, to believe that there is some strength in his own race [Philippa: Yes.], so, in a sense, you know, that moment, for his character, we felt was very important.

Edlothiad's answer gives a great perspective on why Aragorn was presented with this challenge, and why it is an important moment for Aragorn, but I'd like to add my thoughts on why Frodo would choose to test Aragorn in this way.

Frodo needed to see if he could continue with the Fellowship.

I think there was a lot of subtext in the question,

Can you protect me from yourself?

It seems like the unspoken part of that question was, Can I continue with the Fellowship, or will my companions betray me, one-by-one?

Aragorn certainly saw the implication of the question, because he responded in a way that showed his understanding that Frodo could no longer trust the Fellowship and must depart on his own:

I would have gone with you to the end... into the very fires of Mordor.

and later, to Legolas:

Frodo's fate is no longer in our hands.

Frodo was at a crossroads, and he needed affirmation that the ring was his own burden.

Part of Frodo still believed that someone else may be better suited to carry the ring, and he needed to solidify his resolve if he was to continue. By challenging Aragorn, he was, to some extent, asking if it really had to be his own burden after all. Aragorn was, by any standard of Middle-earth, the best person to hold an object of such danger and power, so refusing it proved to Frodo that he could not allow anyone else to possess it.