To answer your specific question: The specific ability of the A350 is to extend the inboard flaps independently of the outboard flaps and to extend them by only a few degrees.
The A350 has a high wing sweep of 35° and a high wing span. The result is, that seen from the direction of flight, the outer parts of the wings lie far behind the inboard flaps. This means, that the center of lift of the wings is actually located around where the inboard flaps are. If the inboard flaps are extended by up to 2° (the maximum possible during cruise flight), the additional lift created by them will be located close to where the center of lift was already before!
I think, this can be seen rather easy from figure 2 of the design paper linked by Speedalive. The blue line and label "inboard flap" shows where those flaps are, that can be slightly extended. Keep in mind, that those flaps are actually not just at the edge of the wing, but start already a meter or so before. If the flap is rotated by up to 2°, it produces additional lift along all of its surface, not only the edge. So the center of additional lift created by the inboard flaps alone might even be slightly before the center of lift of the whole wing.
Furthermore, as already said by others, this slight extension of the flaps by 2° creates more uplift and thus allows to reduce the pitch angle of the whole aircraft by maybe 1°. This reduces the angle of attack on the wings by the same amount, and with the typical asymmetric wings used on aircraft today, lowering the angle of attack usually leads to a slight forward movement of the center of lift.
So, putting it all together: Additional uplift is created around the back of the inner part of the wings. In exchange, the angle of attack is reduced and less uplift is created on the outer part of the wings, which actually lie behind the (slightly) extended flaps, and also on the front parts of the inner wings. Both reductions have a combined center of lift reduction, which is close to the center of the additional uplift from the flap extension.
What others have already pointed out: At the beginning of the cruise flight, the fuselage is usually heavy, as the center tank is still filled with fuel. So more force needs to be transferred from the wings to the fuselage. If the additional force is created by increasing the angle of attack (or by flying at a lower altitude in denser air), it is created along the whole length of the wings and the whole structure needs to be able to carry that additional force over the full length of the wings. With the slight extension of the inboard flaps only, the additional uplift is created on the inner wings instead, close to the fuselage. So the maximum weight, that the outer wings have to carry for extended periods of time, is reduced. They don't need to be so strong, so less material can be used there, saving on aircraft weight.
Trim Tanks
I tend to disagree with others, that claim, that this "cambered wing" is a good replacement for the trim fuel system in A330 and A380. They are different systems to optimize fuel efficiency: The "cambered wing" is about reducing the angle of attack during heavy flight phases. It actually can also be used later, for example, when ATC asks the pilot to climb 1000 feet higher than what would be optimal for the current weight to avoid other traffic. Then again, flying higher, the aircraft would either had to increase the angle of attack and thus also the drag, or use the slight extension of the innermost flaps as a more efficient alternative!
Trim fuel, on the opposite, is about managing the center of gravity and let it get closer to the center of lift. When flying close to ground during initial climb and final approach, a large separation between these two points is essential for stable flight, especially during upsets like crosswinds and similar. During cruise flight, less separation is required and trim tanks can help to reduce it. During take off and landing, even more separation is required, and this is achieved by extending all flaps (not only the innermost ones) by a large angle (and not only by 2°), which substantially moves the center of uplift backwards.
On the other hand, the high wing sweep of the A350 allows also some trimming by exploiting the different fuel tanks of the wing only: Pumping fuel to the wings' outward tanks (or just consuming the fuel from the inner tanks, first) will move the center of gravity substantially to the back.