Its easiest to understand the difference when both the larger and smaller sensor have the same megapixels. If we have a couple hypothetical cameras, one with a smaller APS-C sensor and one with a Full Frame sensor, and assume both have 8 megapixels, the difference boils down to pixel density.
An APS-C sensor is about 24x15mm, while a Full Frame (FF) sensor is 36x24mm. In terms of area, the APS-C sensor is about 360mm^2, and the FF is 864mm^2. Now, calculating the actual area of a sensor that is functional pixels can be rather complex from a real-world standpoint, so we will assume ideal sensors for the time being, wherein the total surface area of the sensor is dedicated to functional pixels, assume that those pixels are used as efficiently as possible, and assume all other factors affecting light (such as focal length, aperture, etc.) are equivalent. Given that, and given that our hypothetical cameras are both 8mp, then its clear that the size of each pixel for the APS-C sensor is smaller than the size of each pixel for the FF sensor. In exact terms:
360mm^2 / 8,000,000px = 0.000045mm^2/px
-> 0.000045 mm^2 * (1000 µm / mm)^2 = 45µm^2 (square microns)
-> sqrt(45µm^2) = 6.7µm
864mm^2 / 8,000,000px = 0.000108mm^2/px
-> 0.000108 mm^2 * (1000 µm / mm)^2 = 108µm^2 (microns)
-> sqrt(108µm^2) = 10.4µm
In simpler, normalized terms of "pixel size", or the width or height of each pixel (commonly quoted on photo gear web sites), we have:
APS-C Pixel Size = 6.7µm pixel
FF Pixel Size = 10.4µm pixel
In terms of pixel size, a FF 8mp camera has 1.55x larger pixels than an APS-C 8mp camera. A one-dimensional difference in pixel size does not tell the whole story, however. Pixels have two-dimensional area over which they gather light, so taking the difference between the area of each FF pixel vs. each APS-C pixel tells the whole story:
108µm^2 / 45µm^2 = 2.4
An (idealized) FF camera has 2.4x, or about 1 stop worth, the light gathering power of an (idealized) APS-C camera! That is why a larger sensor is more beneficial when shooting in low light...they simply have greater light gathering power over any given timeframe.
In alternative terms, a larger pixel is capable of capturing more photon hits than a smaller pixel in any given timeframe (my meaning of 'sensitivity').
Now, the example and computations above all assume "idealized" sensors, or sensors that are perfectly efficient. Real-world sensors are not idealized, nor are they as easy to compare in an apples-to-apples fashion. Real-world sensors don't utilize every single pixel etched into their surface at maximum efficiency, more expensive sensors tend to have more advanced "technology" built into them, such as microlenses that help gather even more light, smaller non-functional gaps between each pixel, backlit wiring fabrication that moves column/row activate and read wiring below the photo-sensitive elements (while normal designs leave that wiring above (and interfering with) the photo-sensitive elements), etc. Additionally, full-frame sensors often have higher megapixel counts than smaller sensors, complicating matters even more.
A real-world example of two actual sensors might be to compare the Canon 7D APS-C sensor with the Canon 5D Mark II FF sensor. The 7D sensor is 18mp, while the 5D sensor is 21.1mp. Most sensors are rated in rough megapixels, and usually have a bit more than their marketed number, as many border pixels are used for calibration purposes, obstructed by sensor filter mechanics, etc. So we'll assume that 18mp and 21.1mp are real-world pixel counts. The difference in light-gathering power of these two current and modern sensors is:
7D APS-C: 360mm^2 / 18,000,000px * 1,000,000 = 20µm^2/px
5DMII FF: 864mm^2 / 21,100,000px * 1,000,000 = 40.947 ~= 41µm^2/px
41µm^2 / 20µm^2 = 2.05 ~= 2
The Canon 5D MkII Full-Frame camera has about 2x the light gathering power of the 7D APS-C camera. That would translate into about one stops worth of additional native sensitivity. (In reality, the 5DII and 7D both have a maximum native ISO of 6400, however the 7D is quite a bit noisier than the 5DII at both 3200 and 6400, and only really seems to normalize at about ISO 800. See: http://the-digital-picture.com/Reviews/Canon-EOS-7D-Digital-SLR-Camera-Review.aspx) In contrast, an 18mp FF sensor would have about 1.17x the light gathering power of the 21.1mp FF sensor of the 5D MkII, since fewer pixels are spread out over the same (and larger than APS-C) area.