A digital image sensor is ultimately a device that uses the photovoltaic or photoconductive properties of photodiodes to convert photons into electrons (charge), which can later be read out as a saturation value and converted to a digital pixel. This is an analog-to-analog then analog-to-digital conversion process.
The key behavior of a photodiode relevant to imaging, converting photons to electrons, improves with total surface area. The more surface area, the greater the area to detect photon strikes per photodiode, and the greater the physical material area within which electron charge (signal) can be collected. In other words, larger physical pixel area equates to higher signal ratio. The "depth" of a well is ultimately immaterial to modern Bayer-type CFA sensors, as deeper penetration only really has a filtration effect...the deeper a photon penetrates a photodiode, the more blueshifted light will be filtered out in favor of redshifted light. This is due to the response curve of the type of silicon used in photodiodes...which are more sensitive to infrared wavelengths than visible light wavelengths, and more sensitive to visible light wavelengths than ultraviolet and x-ray wavelengths.
Finally, being electronic devices, image sensors produce a variety of forms of electronic noise. In particular, they are susceptible to a low number of electrons in any given photodiode being generated from the low level of dark current that is always running through the sensor. Being devices sensitive to electromagnetic frequencies, the intrinsic field of the sensor itself can be affected by strong, nearby devices that emit electromagnetic frequencies of their own (if its not shielded properly) which can produce banding. Slight differences in the exact electrical response of each pixel can produce slight variations in how the charge accumulated in a photodiode is read out, and there can be thermally induced forms of noise. These forms of noise create a signal floor wherein it becomes difficult or impossible to determine of a digital pixel level is the product of an actual photon capture or due to electronic and thermal noise. So long as the image signal is larger than the noise floor, or in other words the signal to noise ratio (SNR) is high, a useful image can be produced.
All things being equal...and by that, I mean the same number of pixels, the same ultimate sensor design characteristics, the same amount of dark current, the same amount of read noise...a smaller sensor will be noisier than a larger sensor because the larger sensor, with the exact same number of pixels, can have larger surface area for each of those pixels, allowing more electrons to be captured for any given photon strike. A larger pixel has a higher maximum saturation point, which allows more total electron charge before the pixel is "full" or totally white. That intrinsically increases SNR, reducing the impact that electronic noise has on the final image signal, producing less noisy images at exactly the same settings as a smaller sensor.
Dynamic range is the total usable tonal range available from a sensor. It is ultimately affected by the amount of electronic noise present in a sensor and the maximum saturation point of the pixels, or the ratio between the mean of electronic noise and the maximum saturation point of each pixel in the sensor. Again, all things being equal, dynamic range will be better on a larger sensor as the SNR, even at low signal levels, is going to be slightly better than a smaller sensor, and at higher signal levels it can be significantly better. As is often the case with image sensors these days, increasing pixel size, or for that matter increasing a pixels maximum sensitivity (ISO), has the effect of also increasing the maximum amount of read noise at low ISO. This is ultimately due to poor control over electronic noise to start with, resulting in higher read noise at minimum ISO for larger sensors than for smaller sensors. While the increase in read noise is often still minor compared to the increase in maximum saturation point, and therefor does not affect maximum SNR much, it can mitigate or eliminate any gains at the sensors minimal SNR level, reducing or eliminating any improvement in dynamic range as well.
