Why did Frodo need to go it alone, leaving Aragorn and Legolas behind?

Frodo saw the effect the ring had on Boromir (Sean Bean), and knew that eventually it would corrupt the rest of the Fellowship. No one is immune to the ring. Even, Gandalf and Galadriel refused to touch it. Frodo as a hobbit, is a bit resistant to the ring, but not completely impervious to it's influence. This is something that Lady Galadriel had foretold (emphasis mine).

Galadriel: It is what will come to pass, If you should fail. The
Fellowship is breaking, it has already begun. He will try to take the
Ring, you know of whom I speak. One by one, it will destroy them.

So Frodo knew that if he was to have a real chance at destroying the ring he would need to do this by himself and carry the burden alone. Of course Sam had other ideas.

Galadriel: You are a Ring-bearer, Frodo. To bear a Ring of Power is to be alone.