I already understand all the theory behind how a bigger sensor leads to better control of depth of field, and lower noise. But I've yet to find a place that can explain why a bigger sensor gives you a bigger dynamic range?
The size of the sensor does not matter, it is the size of the pixel. Having that said, bigger sensors like in full frame cameras tend to have bigger pixels.
You can estimate the size of the pixel by taking the size of the sensor and divide it by the number of pixels. This calculation is not accurate because most sensors have gaps between the pixels and these gaps differ in size. That is why I'm saying "estimate".
Now, think of a pixel in the sensor as a box and photons as balls. The bigger the box, the more balls it can contain.
Assuming we have box A and B. Box A can contain 256 balls and box B can contain 512 balls. Now lets arrange a box matrix of A type and throw a lot of balls in the air. We want gather some statistics of where the balls fell.
In the middle one of the boxes contains 256 balls and in the edges the boxes contains ~20 balls. We cannot know if in the middle only 256 balls fell or more. Our measure is limited to maximum for 256 balls.
Now lets repeat this experiment but now with boxes of type B. Now we can see that in the middle the box contains 347 balls and on the edges the boxes contains ~20 balls.
Our measure is far more accurate. That's exactly what happens with the photons hitting the sensor. Bigger surface can contain more photons and can measure a larger dynamic range. In our example the dynamic range is twice as big in the bigger box.
If the pixel is full of photons the translation to color will be a fully saturated color but with a larger pixel surface we will get a better result, thus, improved dynamic range.
Here is a picture that can demonstrate my explanation:
For deeper explanation you may want to check this article:
There is no reason that a bigger sensor can give a bigger dynamic range or lower noise other than more surface area per pixel, however full frame cameras tend to be higher end units and therefore tend to have better sensors.
Note that there is no reason that a lower resolution, smaller sensor couldn't have better noise and dynamic range performance if made at a similar quality to a full frame sensor. The number of pixels per inch on the sensors surface and the quality of the sensor are the bits that matter.
Considering ideal sensors where photon noise is the only concern then the larger the sensor the greater the dynamic range. Dynamic range is the difference between the point at which the sensor becomes saturated and the point at which any detail is lost to noise in the shadows.
A larger sensor will either have larger pixels, or more pixels. Larger pixels mean a greater capacity to store charge (all else being equal) and more light being captured per pixel hence less light in the shadows, hence greater dynamic range. More pixels means the similar noise per pixel but more pixels to average over to reduce shadow noise, and hence increase dynamic range.
In reality there are other sources of noise, namely read noise, where the analogue signal produced by the photosites picks up noise prior to being digitized. This can have an effect on dynamic range that is far stronger than differences in sensor size. Low intensity signals from dark areas of the image are particularly sensitive to read noise hence the large impact on DR.
New technology (shortening the path from sensor to ADC, sending the signal twice and comparing the results) can virtually eliminate read noise this enables APS-C sensors such as the Sony Exmor to exceed the dynamic range of Canon's 2.5 times larger full frame sensors by almost an order of magnitude!
It is also necessary to distinguish between dynamic range in good light and dynamic range in poor light. The former is determined mainly by read noise, hence a small sensor can excel provided it has low read noise and a decent enough well depth. The latter is dominated by photon noise (raising the ISO in low light amplifies photon noise but not read noise) hence larger sensors tend to perform better here. Again not every sensor follows the trend.