I think this falls over the edge of my understanding, but here goes. Keep in mind that this is the first time I've ever actually tried to make sense of redshifting and wavelengths, so it's very possible I've gotten something totally wrong or backwards.
Redshift isn't a property of the photon itself. The photon doesn't age or turn red as it travels. Redshifting is a property of one object travelling away from another object, and the wave (not particle) of light between the two. If the two objects are static in relation to each other, there is no redshift, regardless of the time or distance.
In our universe, however, on a intergalactic scale, everything is moving away from everything, in all directions. We are all dots on a balloon, and the balloon is inflating. (If this gets you going about curvature, then it's a flat piece of balloon being pulled from the perimeter). Because the expansion is speeding up, older things are moving faster, and are more redshifted.
Since time = distance, and time = speed, therefore further = older = faster = more redshift.
Light is photons, which are particles, but it is also a wave. The wavelength is the distance over which the wave's shape repeats. Imagine that there's a line between two objects, and that line has a repeating wave pattern. Let's say that there are....30 waves between Object A and Object B. The length between each wave equals X. If the objects are static, X is unchanged. If the objects begin to move together,the waves get tighter and X becomes shorter (X-, aka blueshift). If the objects move apart, X becomes longer (X+, aka redshift). The distance between waves is increasing, but the quantity of waves remains the same. There's still 30 waves, but they have to stretch to accommodate the increasing distance. That's redshift. The wavelength (distance between the same point on two adjoining waves) of light increases as objects move apart.