Of course, this is just my opinion, but I think that Aragorn was very sad to see that the enemy had Frodo's mythril shirt. For it to have come to them may have meant Frodo's death, torture or something else very unpleasant. Aragorn's love for Frodo and his respect for the little hobbit walking into Mordor showed in his tears and quavering voice at the "For Frodo" moment. Aragorn running headlong into Mordor shows his courage and committment to a cause despite the overwhelming odds against him.
If you will also remember that he did say something, not in these exact words but, I will not believe it, I think that is also a key point in all the things that happen between then and the battle.