Meaning of "to believe the impossible"

Solution 1:

"To believe the impossible" to me sounds truncated. The phrase is familiar, true, but when thinking about it, I would rather see:

• To believe in the impossible, or
• To believe the impossible could be possible
• To believe in that which is impossible

In its -- to me -- truncated form, it seems to imply that "the impossible" is an entity of its own, e.g. to believe the rumours.

Therefore I would interpret the sentence in question as:

To believe that things which most people consider impossible, is actually possible

Which is different from your options in that it uses a subjective viewpoint. To believe the impossible to me feels more like the viewpoint of a free thinker, a radical, or whatever adjective you prefer for someone who is not satisfied with mainstream opinions (or generally accepted facts).

Solution 2:

I'm getting the impression that it's used when you're asking someone to do a leap of faith and believe a story that person is unlikely to believe if it wasn't for you begging him to.

Solution 3:

As completely anecdotal evidence from my own experiences, believing the impossible refers to the action of deciding to believe in something which seems, at first blush, to be impossible.

Since our achievements can be limited by our aspirations, this phrase is usually said to inspire greater aspirations in the hearer.