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:

Dynamic_Range