Why can we see through rain? I am gazing through my office window into a heavy rain. I am thinking that raindrops are like small lenses that bend the light. Thus I am surprised, that I can clearly see other buildings through the window. 
So, why is it that we can see through the rain? Is the density of raindrops simply too low?
 A: Many of the photons coming from nearby objects will travel to your eye without striking a rain drop. However, photons traveling from more distant objects have a greater chance of hitting a rain drop before reaching you. This makes more distant objects seem dimmer or more difficult to see.
A: Since there is space between the rain drops, some rays of light travel from the building to your eye without hitting a rain drop. Some other rays if light get scattered randomly when they hit rain drops. Your brain sees the building through the haze because the unscattered light always conveys the same image of the building while the scattered light is random. Your brain knows what's real and what's just rain scattering because what's real is constantly there.
A: You may be interested in this paper by Garg and Nayar which analyzes the visual distortions and other effects produced by rain.  The purpose of the paper was to be able to edit rain out of visual detection systems, and to be able to edit rain into computer graphics.  Scroll down to section 4.1 Dynamics of Rain, and Figure 5 which shows how individual raindrops produce "complex mapping of the environmental radiance".
The authors conclude that because "rain drops fall more or less in the same direction...the distribution of drops is uniform over space and time, [and] the binary field...due to rain is wide sense stationary."
In other words, when you see through rain, you get a generally stationary view of the background, but if you follow the light paths through the raindrop as illustrated in Figure 5, you will notice that each drop directs light not only from the background, but also from above and below, toward you.  It's as though your vision is composed of numerous pixels that include not only what you would see from the direction you are looking, but also from other directions in which your gaze is not directed.  This may explain partially the hazy nature of what we see in a heavy rain.
A: I would dispute that we can see through rain.
Any line-of-sight that ends on a rain drop is blocked from reaching the distant object. As we look deeper into the field of rain, more lines-of-sight end on raindrops and images of distant objects become less distinct. Eventually, when the probability of a LoS ending on a rain-drop approaches unity, the depth of view is limited to that distance.
This is exactly what happens when you are out in a very rainy environment, such as a mountain range or at sea. The limit of sight (visibility in seaman's terms) might be around 1km.
An exact parallel to this occurs in Olber's Paradox
A: Low density of raindrops is the answer. However, the change we see in the scene due to the rain can be explained with modulation transfer function (MTF) (for any optical system) or Contrast Sensitivity Function (CSF, characteristic of the eye only). 
Higher spatial frequencies added to the scene, and lower spatial frequencies suppressed with falling raindrops lower the contrast of the image.

Courtesy
CSF tells us how a contrast of an image in our brain changes with spatial frequencies in the scene:

Courtesy (photopic - well-lit, mesopic - medium, scotopic - low light scene).
Now to the numbers.


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*Average speed of a falling rain drop: 10 m/s

*Human eye integration time (time when an image of the scene is created in the cells of your eye): 0.1 s (rods, shape perception), 0.01 s (cones, color perception). We choose the time for rods: when you look through the rain the first thing you look for is the shapes of the objects (look at distant objects in the picture).


During the integration time the raindrop flies 0.1[s]*10[m/s]=1[meter]. So, roughly speaking, each rain drop is a meter long object in the final image your brain deals with. During this time (0.1 s) rain drop manages to refract ("bend") so many light beams (so many photons flying towards you) from so many directions that taking them all at once gives you almost no useful information about the scene. So we may assume that the raindrop is almost opaque meter-long object in the scene. Almost because:


*

*JDlugosz' comment: "If you are integrating the exposure, the "meter long" rod is not opaque because most of the time any segment is not blocked.  You'll get 90% clear view and 10% noise from the fraction of the exposure time that a raindrop was in that spot."

*Alex's idea about photons coming from another directions is true and raindrops are not totally opaque to us due to diffraction effects.


Alex also noticed correctly that distant objects are harder to see in the rain. As density of meter-long opaque objects grows the contrast of the image decreases. This is due to modulation transfer function) of the eye.
A: Two main reasons. First, the raindrop density is really low. Recall how it may sometimes seem it's pouring rain but you go out and barely get hit with some 10 droplets per second. It makes sense, when it's raining, it's still mostly air. If rainfall is $10\, {\rm mm/h}$ at $10\,{\rm m/s}$, the density of droplets must be the quotient of these fluxes ($\sim 3.6\cdot 10^{-6}$). That is, in heavy rain, only a few parts per million of air is droplets (comparable to cloud density). An important factor here is also projection: a visual image is 2D projection of the droplets: raindrops are huge and so this small percentage of volume is mostly concentrated in a few dots at any given time (+blur helps even more). Fog is worse mainly because it covers your field of view more efficiently: a droplet of volume $V$ covers $V^{2/3}$ of your vision. $N$ droplets of volume $V/N$ cover $N^{1/3} V^{2/3}$. Scattering difference also help to obscure even more efficiently, but mostly it's just fragmentation.
Another very important part is the motion blur. Droplets are so fast that within time resolution of a human eye (let's say a 20-50 Hz, depending on the light conditions), the droplet travels up to a metre distance. So the droplet never fully obscures a certain part of your visual field, it only "blocks" your vision for a fraction of the "exposure time".
That being said, when you are looking through a sufficient amount of rain, it does lower visibility quite a lot. Curtains of rain on the horizon are a common sight (possibly with a rainbow, which is, again, see-through).
