The first thing to take into account is how stress builds in a material. This is rather intuitive and something we all know, but takes a bit of thinking to back it up mathematically. It's quite easy to get sucked into the world of 'Second moment of area' which is incredibly important for calculations, but not strictly necessary for understanding how it works (I hear screams of "heresy"..).
Essentially, when a beam bends, the bottom fibres are stretched and pulled apart (tension), while the top fibres crush together (compression). The beam will fail (e.g break) when either the top fibres reach their maximum compressive strength, or the bottom fibres reach their maximum tensile strength. 333
Simple really! See my vquick sketch below:
So the implications of this is that at some point between the bottom and top fibres, there is a point where there is no stress (where it transitions from tension to compression). The maximum bending moment (engineer talk, not strictly important for understanding!) that a beam can take is limited by the compressive or tensile strength of a beam, and so working backwards we can work out the difference between a beam that is standing upright and one lying flat.
I've done a quick hand calculation with a few assumptions (for simplicity I've assumed the beam will fail in compression and has a compressive strength of 10N/mm^2 and changed the beam from 2"x10" to 20mm x 100mm). Starting with this, you can work out the 'area' of the stress triangle. You then multiply this by the area of the beam that this stress triangle occurs over.
As you can see, the 'taller' of the two beams has a larger stress triangle and thus can resist more bending! This is because more of the beam is located further away from the neutral axis (the point where the beam is neither in compression or tension)! Basically if you were to slowly increase the load, the top fibre of the taller beam would take longer to reach the killer 10N/mm^2 limit that I've given it.
I hope this is clear and simple, it's a little irritating to explain what is a relatively simple concept because there's so much ground to cover.
EDIT: I plan on re doing those sketches/calcs soon, hopefully with some colour. I quickly drew them up at work so didn't put much effort into them! 333