When you accelerate away from me in an elevator, only one of us experiences additional forces during that acceleration. Only your body feels heavier as the elevator speeds up.
Here’s a minute physics video that shows how acceleration explains the twins paradox (the “common” explanation) https://youtu.be/0iJZ_QGMLD0
After you watch minute physics video, you might feel like you understand. Nope, can’t have any of that. Now, you must watch this Fermi lab video that says the common explanation is “not fundamentally correct” https://youtu.be/noaGNuQCW8A
Fermilab video is hard to grasp intuitively to me, but one important thing it did was it cleared that confusing misconception of acceleration, which was induced by [too many] other videos.
After watching these incorrect videos it was still unclear why acceleration even matters, because with 1g accel in space you can reach subluminal speed in a matter of days, and then, at the distant location, turn back with just 2x long 1g again, and then “brake” with 1g again to land on earth. While the twin on earth experienced that 1g all the time. One may even pick a distance and a (probably hyperbolic-y) route with rotations so that all of the journey would consist of a constant 1g for a flying twin, exactly as a sitting one experiences. Put them both into opaque boxes, knock senseless for a couple of hours at launch, and neither of them would even tell who is where, until opened.
I think the most important point if you want an intuitive explanation is that the observer who is in the same place at the beginning as at the end has aged more.
If you were to ask which of the twins is older at the time the second twin reaches the far away star, the answer is that the question doesn't make sense. They are in different places, so there is no way to compute a fixed time.
Imagine each twin is broadcasting a video signal of their face for the entire duration of the trip. The twin on earth puts a big red dot on the screen the moment that the traveling twin had reached the distant star in their frame of reference; while the twin on the ship would start putting a big red dot on their video the moment they reach the star in their own. Because the speed of light is limited, over the whole duration of the departure trip, for both twins, the latest image they see of their sibling will be younger than themselves.
When the twin on the ship reaches the star and starts emitting the red dot, they will have aged L/gamma * v years. By the time the twin on Earth sees this image, they will have aged L/v + L/c years = L(c+v)/c*v years, so they are [L(c+v)/c*v] / (L/gamma * v) = gamma(c+v)/c years older.
The twin on Earth will start emitting the red dot after L/v earth years, or L/gamma*v ship years. This signal will meet the ship after L/(gamma*(c - v)) ship years from the moment it was sent - so in total, L/gamma*v + L/gamma*(c-v) = L*c/gamma*v*(c-v) ship years. So, they are [L*c/gamma*v*(c-v)] / (L/gamma*v) = c/(c-v) ship years older than their sibling, which is gamma*c/(c-v) earth years.
While both see that they are older than their sibling at the time they first see their sibling's red dot, there is an asymmetry coming from the fact that one of them is at the same position as they were initially, while the other one is at a different position, relative to the star. Equivalently, there is an asymmetry that is caused by their different definitions of "the moment the ship reaches the star".
Acceleration isn't relative, velocity is. An object undergoes acceleration when a force is applied to it. Objects "know" when they are being accelerated, just like how you know you're being pulled down by gravity. The same can't be safe for velocity.
