That term conveys a number of advantages for the writers:
Councils are generally democratic (or at least republican), and thus generally viewed more favorably by modern audiences than kings, emperors, etc. A powerful group governed by a "council" gives the impression of at least a semi-enlightened society.
At the same time, the term "High Council" retains a more formal, even ceremonial feel than many modern terms, and that adds gravitas to the story. He's not just a Congressman, he's a "Member of the High Council."
It's also a suitably generic term that allows the writer to establish certain facts (e.g. it's made up of multiple people, decisions are made by vote, etc) without getting bogged down in minutiae (e.g. is it a congressional or parliamentary system?). It's just a Council.
In many cases, it offers an opportunity for palace intrigue. A Council is made up of individuals, and those individuals can disagree, betray, persuade, or replace each other. A enemy monarch must be opposed directly, but an enemy on a Council allows for the possibility of allies with similar power. "If we can just convince Senator Whosawhatsit, we can stop Consul Evilface by finding and dismantling the Apocalyptathing! Only a member of the High Council has that kind of power!"
Because "council" sounds like they're reaching wise decisions (as opposed to decisions that are "popular", which are common in systems like democracy). Essentially, a SciFi race with a "high council" appears to have a better system of government than humans have been able to come up with.
I don't think the "high" part needs explaining, but here goes. An order obviously sounds more authoritative if it is from a "high" council. This is assuming that "high" is interpreted in the sense of rank rather than recreational drug use.