Full moon days enhance the present quality that you carry on that day.

Here is a section from Sadhguru's book "Of Mystics and Mistakes" where he talks about the effects on humans on full moon days/nights:

Now take the volume of this microphone as an example. Suddenly the volume is increased and it starts blaring. The talk is the same, but suddenly it is a little louder and clearer. It is just like this. There was a little madness; you gave it a little extra energy or you increased the flow of energy, and now everything seems to be magnified. It is the same with love, with laughter, with joy, with everything. It is just that on the full moon day the energy is a little higher.

What makes the energy a little higher? For one, there is a certain aesthetic quality about it. Anything that you look at, if it is beautiful, your receptivity to that object suddenly becomes greater, doesn’t it? Anything that you consider ugly, the moment you look at it, your receptivity to that just comes down. So one thing is that the full moon has a certain aesthetic quality; this definitely improves your receptivity. Another thing is that the planet has moved into a certain position in relation with the satellite that makes the vibrations very direct and forceful. You know the tides are rising, because the gravitational pull of the moon is working on the water more than on other days. So water is spilling over and trying to jump up. Similarly your blood is trying to jump up. When the circulation of blood increases in your brain, whatever is your quality gets enhanced. If you are little mad, you become madder. If you are peaceful, you become more peaceful. If you are joyous, you become more joyous. Whatever is your quality, it is just being pushed up. People only noticed madness, because most of the people are in that state, okay? But it is not only madness; it is so about everything on a full moon night.

I don't necessarily recommend focusing on Sadhguru's explanations of why this is happening. I would argue that he is just trying to give you similar examples that are closer to your experience, in order to clarify what he is observing. When Sadhguru uses those examples, those might not align properly with what Sadhguru is saying.

For e.g. right now the current scientific explanation for higher tides during full moon days is not connected to the stronger gravitation pool of the moon itself. It is rather connected to the alignment of the moon and sun on that day. Sun's position is diametrically opposite to moon position in relation to earth (which is in-between). Because both sun and moon have separate gravity effect on earth (sun has 1/3 of pull force of moon), during full moon days, sun assists the moon by pulling bodies of waters to opposite side than the moon, so the total effect on bodies of waters adds up and tides become higher.

You also shouldn't focus on scientific explanations either. All of the explanations are simplified to common folk and not correct, and also alliable to change as science expands their understanding. Gravity is actually not a force at all, but effects of gravity are sill there. Similarly, Sadhguru's explanations on "why full moon affects humans" might not be correct, but it's effects are perceived by him.