The gravitational potential energy of the object is $mgz$, where $z$ is the height of the object above some reference point. So if the object moves downwards by one meter, it loses a fixed amount of energy, which in this particular example is one Joule. The potential energy is associated purely with its position --- wherever the object is, if you move it down by one meter, it will lose one Joule of energy.
Because the object covers the second meter more quickly than the first, it loses this fixed amount of energy more quickly over the second meter than the first. That's all there is to it.
To give an analogy with an elastic band: if you stretch an elastic band, its potential energy will be at some increased level. The potential energy depends on just how stretched it is --- the extension of the band from its equilibrium. When you let the band go, its potential energy is going to decrease. But as you let the band go, the contraction will speed up over time. You would have to watch it in slow-motion, but in the first millisecond the band wouldn't contract as much as in the second millisecond, because in the second millisecond the band will have 'got up to speed', hence will be contracting more quickly, hence will be losing potential energy faster.
I haven't said anything new at all here, but maybe from another perspective the matter seems less magical.
Problems like these often boil down to a clash of definition and intuition --- that is to say, your intuition of what energy is may not match its mathematical definition. I was once asked by a friend the question: how can the Earth just keep attracting things to itself indefinitely; won't it run out of energy? Giving a satisfying answer here is hard --- if we define energy in the appropriate way, it's clear that the Earth won't 'run out of it'.
Similarly, the fact that 'gravity gives the object more energy in the second second than the first' is just a consequence of our definition of energy. I mean, when you watch the object fall at constant acceleration, nothing about the motion seems odd or counter-intuitive. But if we look at the quantity
$$\frac{1}{2} m v^2 $$
and how it changes over time, we'll see that it increases more over the second second than the first. This is just a consequence of the way it's defined!
So gravity is more 'powerful' when the object is moving faster. But 'powerful' here has a precise meaning. Gravity exerts the same force on the object regardless of its speed --- if one were to think of 'power' as 'how much force it can exert' (which is NOT the physical definition) then gravity is just as 'powerful' no matter the speed of the object, which to you seems sensible and intuitive. Gravity only gives the object more energy over the second second than the first as a consequence of the definition of energy.
Hope this helps.
