Why do the elements of science have abbreviation?

Answer 1

We abbreviate the elements because it's faster and takes up less room than writing the names in full. Pretty much the same reason anybody abbreviates anything.

There may be some vestigial traces of alchemical practice there as well, where symbols were used at least partially to keep the uninitiated from understanding alchemical documents, though for the most part scientists actively want for their work to be understood by as many people as possible.