How does "slightly" altering the physical construction of an antenna change its impedance?
why does bending a dipole antenna change its impedance?


This is quite a simplistic answer. An antenna has to "interface" with free space. 

Free space has an impedance defined by \$\sqrt{\dfrac{\mu_0}{\epsilon_0}}\$. 

This turns out to be near-enough 377 ohms when you plug the numbers in. 

It's the same as a transmission line - it has \$\sqrt{\dfrac{L}{C}}\$ that defines its high frequency (usually > 1 MHz) impedance.

So. if you adjust the dimensions of a piece of coax you get a different value for the characteristic impedance. If you adjust the antenna mechanically you are adjusting it's ability to deliver 377 times more E field to H field. This isn't necessarily a killer because all that your electrical circuit sees, is a radiation resistance that is somewhat lower than (say) 73 ohms for a dipole: -



Notice that 73 ohms is when the diploe is 0.5 wavelengths long and that there is no reactive component. Altering the shape of the dipole inevitably means increasing capacitance and effectively shortening the effective length thus, the impedance can alter quite a bit.


  What's the "mental" model of how an antenna's impedance is determined?


It's something I've struggled with for years and in the end I come back to the ratio of E field to H field and how that relates to impedance of free space. If you bend the dipole in on itself (I'm not talking about a folded dipole) then it has more end-to-end capacitance and the E field will reduce. That's how I see it anyway but, for more complex antennas it's a different story.