I have read that "Under all possible proper times between two events, the proper time of the unaccelerated clock is maximal, which is the solution to the twin paradox."
I got that for a clock in it's own inertial frame the proper time is minimal compared to that of other inertial frames, which for me sounds like "moving clocks tick slower". But how do I understand this cited statement? Why is the proper time of an unaccelerated clock maximal compared to accelerated clocks?
Unfortunately physics does not answer to the question "Why do things happen?", but to the question "How do things happen?".
The two principles of SR (special relativity) plus assumptions on the homogeneity and isotropy of space yield the Lorentz transformations from which you can derive the time dilation. If you apply the time dilation as measured by an inertial reference frame to an arbitrary (accelerated) frame you get that the proper time of an unaccelerated clock is maximal.
Simply nature works in that way. If you want to go beyond that understanding you deviate to philosophy or religion.
I doubt that an intuitive understanding is possible. It follows from a version of Pythagoras’ Theorem (search the web for details) that regarding the twin paradox the elapsed proper time of the longer worldline (that of the travelling twin) is shorter than that of the twin staying at home. Note, the twin paradox doesn‘t require acceleration.
