To the best of our knowledge, this can't be done
Stars are like batteries. Just as there's nothing to stop a battery in use from draining, there's nothing to stop a star from dying. Excluding rechargeable batteries, any action taken to rejuvenate a dying battery is ultimately destructive to the battery. Stars are no different.
But, let's run with the idea that we could rejuvenate a star for a moment.
Conceptually, there's only two ways to rejuvenate a star.
Add more fuel
Put the energy back
Add more fuel: Using my battery metaphor, adding more fuel makes the battery bigger. All the old "fuel" is still there and can't be removed. For the battery, adding more fuel makes the battery less efficient, but also makes it bigger such that it no longer fits in its application.
From the perspective of a sun, adding more fuel (mostly hydrogen to fuse into heavier elements), means adding more mass. More mass means a greater liklihood of both explosion and collapse (into a black hole). It also changes the orbital dynamics of the entire system (your world may no longer be habitable...). So, it might buy you a little time, but ultimately everything's worse than before.
You could claim that your society is so advanced that it could remove the old battery material to make space for the new battery material. If you can do that, why not just move your planet to a younger system. You'd need to justify why your race is so dedicated to their corner of space that they'd rather bring the mountain to Muhammad, so to speak.
Put the energy back: This is the rechargeable battery concept. Stars fuse mass and the result is energy. You could put the energy back, which (theoretically) reverses the fusion and rejuvenates the star.1 However, just as too much recharging can cause a battery to leak or explode, if this process were actually possible, it would result in a very unstable condition that would likely result in a spectacular explosion.2
More to the point, however, where are you getting the energy in the first place? You'd need to rob the energy of one star to stuff it into yours. We're back to moving the mountain to Muhammad... if you could do that, why not just move your planet to that other star?
The cost exceeds the benefit
Even if you could do either of the above, what time it buys you is debatable. Remember, you're expending energy just to rejuvenate the star (either by moving mass or by injecting energy). The cost to your society will be larger (much larger) than the cost of simply moving to a new star system. Energy isn't free and is never more efficient than 100%, which means the energy put into rejuvenation is greater than the energy you'll get out of the rejuvenation process.
I freely admit that some Clarkean Magic could be applied that makes opening a portal to N-Space or some such to siphon off a ton of energy in some way that wouldn't cause the collapse of reality as we know it today possible. But, you're asking us how that Clarkean Magic would manifest. Beats me. That's fiction so pure and refined that it's intistinguishable from fantasy.
1 I'm willing to bet you a milkshake that the physicists on this site, if not rendered incapable of typing at this point due to their laughter, will gleefully explain how what I just explained is a nice concept but entirely nonsense. They'd be correct. Stuffing solar wind back into a star is not the same as stuffing electrons back into a rechargeable battery. "Unfusing" a star would make for good fiction, but not particularly good science fiction.
2 A better metaphor would be sticking a small capacitor into your house's electrical outlet. Yes, the capacitor is rechargeable, but the consequence of this experiment is a spectacular explosion. DO NOT DO THIS. You've been warned.