If objects don't move when the universe expands, how can the expansion result in redshift? I am approaching this topic from a layman's perspective and view this as a logic puzzle, but when someone tries to get me to accept the logic, I'm never convinced by their arguments. 

According to the wiki page on the expanding universe, the universe is expanding (which is how we see redshifted light from distant galaxies), however this expansion does not result in the movement of objects across space away from each other. Instead, this expansion is the formation of new space such that all of the objects in space maintain a static relationship with space. 

This should mean, that as light produced by a distant light source that is expanding away from us, it is not moving away from the photon. The photon then travels across space, and this trip takes longer as a result of the expansion, but once the photon is in space near to Earth, the earth is not moving away from the light. By the time the light is at the same distance from the telescope as a flashlight held in front of the telescope, the telescope is not moving away from the photon any more than it is moving away from the photon produced by the flashlight. 

So, IF the expansion is not the result of motion across space, HOW do we show redshift?????? 

Another comment, laymans perspective
The whole reason it can make sense that objects far enough away can expand away from us FASTER than the speed of light (which is the reason why telescopes can't see further) is that these objects are not moving across space to achieve this expansion. 

Another comment in reference to the Raisin Cake. 
understanding the expanding universe as a raisin cake swelling, therefore the actual cake that is in contact with any raisin stays in contact with that raisin,, the raisins do not move through the cake,,, APART. BUT, in space, the raisins actually ARE moving through the cake. Also, the PHOTON that travels across space from the distant light source to the telescope, travels across space. additionally, it's NOT useful to think of this transaction of energy in a raisin cake, because the raisins DO travel across space away from each-other as the cake swells, but this is NOT how we can possibly understand objects in space expanding away from us on Earth faster than the speed of light.
The redshift is indeed not caused by the movement of the objects, but by the expansion itself. This is a theoretical result from considering the FLRW metric for a light ray traveling on a null geodesic.

A derivation can be found here.

The cosmological redshift $z$ is given by $z+1 = 1/a$, where $a$ is the scale factor (the size of the Universe) at the time the light was emitted. Notice that this result doesn't care how the Universe expanded. In principle, you could imagine a universe which was static when the light was emitted, and static when the light is observed, but expanded by the factor $a$ at some point in between, and you'd observe the same redshift as if that universe had expanded gradually in the meantime. This is in sharp contrast to a normal Doppler shift, where in such a hypothetical case, the would be no observed redshift.

Note also that galaxies receding faster than the speed of light $c$ is no hindrance for us to see them; all galaxies with a observed redshift larger than $z\simeq1.4$ recede faster than $c$. The most distant observed galaxy GN-z11 has a redshift of $z=11.09$, and recedes at $v\simeq2.2c$. When it emitted the light we see, it receded even faster, namely at $v \simeq 4.3 c$. See the ant on a rubber rope "paradox" for an intuitive explanation.