I'd say it's first and foremost a political tool. Pseudo dictators especially love it.
Let's say you're a mayor who needs to increase the price of a public utility by 10%, what do you do? You announce that this price will be increased by 30%. People get furious, protests ensue (sometimes, it's even desirable if riots start) against such an outrageous increase! What to do? You should be as invisible as possible during this turbulent period and let others take all the heat, only to come out at the end announcing that yes, the people have spoken, and as a result the increase will be limited to 10%. Now, the people are happy; it's democracy in action, they have the power! The mayor is happy; he's got his increase happen. Happy ending.
https://en.wikipedia.org/wiki/Anchoring_effect