Photography is, at its most fundamental, capturing light. Generally speaking, the more of it you can collect, the more useful signal you have, which means cleaner, less noisy photos. When it is said that a bigger sensor can gather more light, it’s not only because of the bigger sensor. In order to keep the same f-number, shutter speed, and ISO, the bigger sensor will require a bigger entrance pupil for the same field of view. (You can think of the entrance pupil as how big the physical aperture will be "seen" by the incoming light.) The larger entrance pupil is what will actually let more light in. Therefore, it’s a function of both the bigger sensor and a larger entrance pupil to match it.

If you have a larger sensor but also a higher f-number (e.g. f/2 vs. f/1.7), you will need to raise the sensitivity (ISO) in order to match the exposure. Bumping up the ISO also means more noise, which basically cancels out the advantage of the bigger sensor. So, for a given field of view, a bigger entrance pupil helps lower noise. A bigger sensor by itself doesn’t necessarily, it’s just that with a larger sensor you should get a bigger entrance pupil for the same field of view and f-number.

As luck would have it, all this is directly related to current competition for the Pixel phones, because the Galaxy S7 and S7 Edge have a smaller sensor, but a lower f-number. The Samsung phones have a 4.20mm f/1.7 lens, and a similar field of view as the Pixels, which use a 4.67mm f/2.0 lens. What this ultimately translates to is that the S7 system should collect a bit more light overall than the Pixels’ system (2.47mm vs. 2.34mm entrance pupils respectively). In other words, claims that the bigger sensor of the Pixel phones should yield lower noise are negated by the bigger entrance pupil of the S7 phones. (To be clear, this is not because of the lower f-number of the S7; if its sensor was even smaller and the lens still was f/1.7, the Pixel could gather more light even at f/2.0.)