Why do people say “I have a quick question”?

[Remedios]

I dont think I've ever been bothered by "I have a quick question" intro, even if answer may not be short and simple. 

The only phrase that drives me completely nutty is "I know this is a dumb question", or conversely "There is no such a thing as dumb question"

Yes, there is. 

[Wrigley2172]

To piggie back off Brian: humans are social creatures and we learn what paths we need to take to get the result we want in the quickest way. Whether they be literal paths to get to a destination or a path in a conversation. Also, there is a social pressure to not bother someone, thus when you say "i have a quick question" the implication is that it wont be a long task that will burden you regardless of what the actual question is. Also i believe its an american cultral thing. We as a society are so busy running around we want the information yesterday. Id be interested to see if someone in the U.K or elsewhere would get someones attention in a similar way

[wasCB14]

Building on what others have said, it steers you into a position where you must:

1. Refuse a small favor (only a jerk would refuse) or

2. Walk away from a half-completed task once the full scope is revealed (how silly to abandon something you've already put time into...and foolish you wasted your time) or

3. Help them with something big

More of a subconscious habit than anything calculated.