An answer in one word: abstraction. Algebra is a part of mathematics where acquiring these elusive skills of abstract thinking is perhaps easier and more natural than in other areas.
Why abstract thinking? I can quote Donald Knuth, the great mathematician, computer scientist, and programmer; he spoke about programming, but these words apply to mathematics as well:
“The psychological profiling [of a programmer] is mostly the ability to shift levels of abstraction, from low level to high level. To see something in the small and to see something in the large.”
Nowadays, the focus of applications of mathematics is shifting from modeling the physical world to analysis of the artificial world created around us by information technology. Understanding of this brave new world is impossible without “ability to shift levels of abstraction”.