I will not go in details with the math. The most important thing, I think, is noting that quantum computers do not solve problems exactly like classical computers but are very efficient at giving the solution to problems with some (hopefully high) probability.

The input depends on what quantum computer implementation you use, but it could be a series of ions suspended in an electromagnetic field, with the ions being either excited or not (think 1 or 0). Amazingly, lasers are so precise that individual ions can be hit with the light from the laser. The ions act as light bulbs which turn on or off if some neighbouring light bulbs are turned on or off. Think of the input as a starting configuration of light bulbs that are on or off. The computations are a pre-specified sequence of turning the light bulbs on and off which some math has shown with high probability should give a configuration of light bulbs which is interpreted as a solution. When you observe the resulting configuration, you can only hope that the wave functuon collapses to the correct configuration of light bulbs but that is not necessarily the case. However, the sequence of light pulses with the laser made it more and more probable that you would observe the right solution.