This is called a super-elastic collision. It is not physically impossible - in a sense, it's just the reverse of an inelastic collision, and by following that reasoning, you can then easily come to understand how it's possible. Here's how it goes.
- In an inelastic collision, the energy after collision is lower than the energy before. You still have a conservation of energy problem, just in the "other direction". But of course we know that conservation of energy is still valid, even though these kinds of collisions exist in great abundance in our world. So there's got to be an answer.
The answer? The "disappearing" energy goes into internal energy in the objects (or to some external, non-kinetic sink, e.g. sound waves.). Mostly, this is in the form of heat, causing a slight elevation of the temperature.
Hence, what must a super-elastic collision involve? Logically, by simply turning this around, it means some internal energy must "go out" and become external energy, i.e. kinetic energy. (Or equivalently in mathematical terms, the objects "absorb" negative energy during the collision, which is equivalent to turning the direction of energy transfer around, just as me giving you negative money means you paying me positive money.)
Now, of course, because of the second law of thermodynamics (the law of increasing entropy), it is "exceedingly unlikely" (in a sense that can be made very precise) for the energy source here to be the same kind of energy - i.e. internal thermal energy - where the energy goes following an inelastic collision. Instead, for it to happen within the lifetime of any being within the observable universe, some sort of low-entropy energy must be present within the objects prior to collision that is ripe for conversion to mechanical energy.
One example may be to imagine the balls' surfaces coated with a thin layer of mild explosive for which the initial impact force is just enough to trigger the chemical reactions that release the explosive energy, while also not being powerful enough to outright shatter the balls so that the analysis is still meaningful, and also that its contribution to the masses of the balls is negligible, so that when it cooks off the masses can be treated as the same after collision. But there's many other ways to arrange it, too: it doesn't matter how, just as long as the relevant conversion of the internal energy is possible.
Or, in short: the objects are "powered" in some fashion.
Of course, this is just the practical way. If we want to seriously entertain the profound way, then we can return to my comment regarding the second law of thermodynamics, and imagine the exact time-reversal, even all the way down to the microscopic level, of ordinary balls undergoing inelastic collision. While we can't realistically arrange for this (see "extremely unlikely"), it is not prohibited by the laws of physics of our world as far as we understand them, because they are time-symmetric, so the reverse evolution will be generated by the same laws as the forward one. In this case, when the balls come back together, the thermal jiggling of their atoms will all happen to conspire by sheer happenstance "just right" so as to give them a "kick" from that internal thermal energy at that point of collision, that is enough to send them out with greater kinetic energy than when they came in (and cooling them down proportionately by the needed amount). Again, it's still conversion of an internal reservoir ... just now the one that "isn't supposed to" happen.