Why am I always compelled to begin a response with "Well, "?

Well, it is a context marker, showing that you are launching a story. Examples of context:

"Did you do your homework today?", "Yes" is an acknowledgment of dialog.

"Did you do your homework today?", "Well, yes" means you want to shift to a story.

"Did you do your homework today?", "Uh, yes", another context marker, means that your answer is unofficial, off-the-record, or unverifiable.

"Well" is not necessarily a needless word. It's a discourse marker. Wikipedia on discourse markers.

Because it gives you time to think. Another common strategy is to prefix your response with:

That's a good question, ...

Yes, we often write like we speak. It will always polish your English if you go back again and do as Strunk and White recommend: "Omit needless words."

It's kind of funny to see you asking us what you are thinking.

My guess would be that you keep doing it because you innately feel like that word actually does impart some important information.

My second guess would be that the information in question is something along the lines of "The words that follow are just musings off the top of my head on the issue you (the other party to which you are responding) just brought up." Without the "Well" it could be taken as some kind of prepared or dogmatic statement, and/or might not nessecarily relate in any way to the previous statement you actually want it related to.