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The key question is: why is this not symmetrical?
One answer is: because one of the twin accelerates, and therefore changes inertial frames.
But the better answer is: understand what happens when the stationary twin sends light signals at constant time intervals to each other. When does the travelling twin receives them?
By doing that, we see that "all the extra aging happens immediately when the twin turns around":
  • on the out trip, both twins receive signals at constant intervals
  • when the moving twin turns around and starts to accelerate through different inertial frames, shit happens:
    • the moving twin suddenly notices that the rate of signals from the stationary twin increased. They are getting older faster than us!
    • the stationary twin suddenly notices that the rate of signals from the moving twin decreased. They are getting older slower than us!
  • then when the moving twin reaches the return velocity, both see constant signal rates once again
Figure 1. Twin paradox illustration with twins sending light signals at regular intervals. Source.
Another way of understanding it is: you have to make all calculations on a single inertial frame for the entire trip.
Supposing the sibling quickly accelerates out (or magically starts moving at constant speed), travels at constant speed, and quickly accelerates back, and travels at constant speed setup, there are three frames that seem reasonable:
  • the frame of the non-accelerating sibling
  • the outgoing trip of the accelerating sibling
  • the return trip of the accelerating sibling
If you do that, all three calculations give the exact same result, which is reassuring.
Another way to understand it is to do explicit integrations of the acceleration: This is the least insightful however :-)