Why does image noise happen? Why are digital images shot with a high iso (which means the camera sensor is much more sensitive to light) so much grainer than pictures taken with a lower iso?  Here is an example of the type of grain I'm talking about.
At first I thought this was a limit on the cameras precision, but the same thing happens to your eyes when it's dark out— everything gets a grainy look to it.  Why is this happening?  Is this a property of light itself?
 A: 
Why are digital images shot with a high iso (which means the camera sensor is much more sensitive to light) so much grainer than pictures taken with a lower iso? Here is an example of the type of grain I'm talking about. At first I thought this was a limit on the cameras precision,

The ISO on a digital camera is the amplification of the signal (and noise). If lighting conditions are bright you would be unable to use a high ISO setting because your image would be white. With a brighter scene and a lower ISO number you are amplifying signal and noise less; the desired image is brighter and the noise is not amplified.

... but the same thing happens to your eyes when it's dark out— everything gets a grainy look to it. Why is this happening? Is this a property of light itself?

It's not a property of light, or the same thing.
The graininess of your vision in low light is what happens when even areas of light or dark, static-like fluctuations, are averaged out into a definite shape, colour or tone; but with the lack of color (photopic) vision.
The "graininess" is much less noticeable when there's plenty of light because your brain has a lot more information to go on, to discern the shapes, movement, colour and tones that our vision is tuned (in our brains) to pay attention to.
A combination of scotopic and mesopic vision affects visual acuity and is called the Purkinje shift.
A more technical explanation is offered in: "Mechanism for selective synaptic wiring of rod photoreceptors into the retinal circuitry and its role in vision (2nd source) (Sept 23 2015), by Yan Cao, Ignacio Sarria, Katherine E. Fehlhaber, Naomi Kamasawa, Cesare Orlandi, Kiely N. James, Jennifer L. Hazen, Matthew R. Gardner, Michael Farzan, Amy Lee, Sheila Baker, Kristin Baldwin, Alapakkam P. Sampath, Kirill A. Martemyanov, where they explain:

"Rods are highly sensitive and able to respond to single photons, thus setting absolute visual threshold. Cones are less sensitive but can respond to a broad range of light intensities of specific wavelengths and thus are essential for daytime and color vision (Kefalov, 2012; Korenbrot, 2012). The functional differences between rod and cone photoresponses are carried downstream through their selective connectivity with distinct classes of bipolar cells (BCs) forming established circuits with known properties that are conserved across vertebrate species (Ghosh et al., 2004; Lamb, 2013; Pahlberg and Sampath, 2011).
In the mammalian retina rods establish synapses with a single class of BC, the rod ON-bipolar cell (ON-RBC), forming the highly-sensitive rod bipolar (primary) pathway (Dacheux and Raviola, 1986; DeVries and Baylor, 1993). In contrast, axonal terminals of cones make synapses with several classes of cone ON-bipolar cells (ON-CBCs) and OFF-bipolar cells (Ghosh et al., 2004) that express different types of postsynaptic glutamate receptors. The contacts for rods and cones are formed at stereotyped positions within close proximity of one another (Mumm et al., 2005; Sanes and Yamagata, 2009), and the connectivity of cones with many classes of cone bipolar cells (CBCs) is thought to play an essential role in contrast sensitivity and temporal tuning.
Similarly, the exclusive connection of rods with ON-RBCs provides a dedicated channel for the high gain transmission of single-photon responses at low light intensities and is indispensable for scotopic vision (Okawa and Sampath, 2007). However, the molecular mechanisms that mediate wiring of photoreceptors with downstream ON-BC neurons and the molecular basis for this remarkable synaptic selectivity are entirely unknown.

Possibly of some interest is this blog post: "Is Moonlight Blue?" by Adam Sydney where he discusses how painters depict low light conditions in their paintings, and says: "Saad M. Khan and Sumanta N. Pattanaik of University of Central Florida have proposed that the blue color is a perceptual illusion, caused by a spillover of neural activity from the rods to the adjacent cones. A small synaptic bridge between the active rods and the inactive cones touches off the blue receptors in the cones ..." - linking to the article: "Modelling blue shift in moonlit scenes using rod cone interaction" (2nd source .PDF) (Journal of Vision August 2004, Vol.4, 316. doi:10.1167/4.8.316) by Saad M. Khan and Sumanta N. Pattanaik.
It's important to point out that other scientific studies such as: "Intraindividual comparison of color contrast sensitivity in patients with clear and blue-light-filtering intraocular lenses" (May 2008) by
Gerald Schmidinger, Rupert Menapace, and Stefan Pieh, and "Effects of blue light-filtering intraocular lenses on the macula, contrast sensitivity, and color vision after a long-term follow-up" (Dec 2011), by Kara-Junior N, Espindola RF, Gomes BA, Ventura B, Smadja D, and Santhiago MR, find no contrast improvement by blocking blue light.
A: On the other hand, there are some forms of noise that are a property of the light.
For instance, if you have a detector that is just counting photons per pixel, and during the detection time you average $N$ photons, counting statistics says that number will vary by $\sqrt N$. (For Poisson distributed photons--there are other distributions). If $N$ is too small, you have to integrate longer to shrink $\sqrt N/N$. The noise is a property of the light because the light comes on random bunches (photons).
Another case is coherent light (sound), such as an imaging radar (sonogram). The signal is a sum over many random scattering centers with random phases. In the approximation that they are all equal, the final signal is effectively a random walk of complex phasors. The total amplitude is Rayleigh distributed, so that the intensity is exponentially distributed. That means, where the intensity "should" be distributed as
$$ I = \delta(I-I_0) $$
with mean $I_0$ and standard deviation $0$,
you measure:
$$ I = exp{(-I/I_0 )}$$
with mean $I_0$ and standard deviation $\sqrt{I_0}$. Note that this noise (called speckle noise) is multiplicative--it's a property of the signal such that you cannot reduce it by increasing the signal power. This noise is a result of the light (or sound) being coherent, and as such is a property of the light.
