"VFR Flight Not Recommended" is a judgement call made by the flight briefer. It means, as you noted from the AIM, that the briefer doesn't think flying around VFR is a smart idea based on what they're seeing.
The usual reasons for a briefer using this phrase are:
- The weather is currently below VFR minimums somewhere along your intended route.
- The weather is currently "Marginal VFR" somewhere along your intended route.
- The weather is forecast to be marginal VFR or below somewhere along your intended route.
- Some other adverse condition (like mountain obscuration) exists along your route or in the area you intend to fly.
The briefer's recommendation is just that: a recommendation -- it is not binding. How you evaluate the flight to make a Go/No-Go decision when the briefer provides this recommendation is your choice as pilot in command. You will need to consider why the briefer is telling you VFR flight is not recommended, and apply your own knowledge and judgment.
A couple of examples:
If the briefer is telling you this because Airmet Sierra is active for mountain obscuration and you intend to fly through some mountain passes prudence suggests that proceeding VFR into mountains that may be obscured by clouds would be a Bad Idea.
If the briefer is telling you this because the weather is currently marginal VFR, but it's forecast to improve and you only plan to shoot a few touch-and-go landings at your home airport you might proceed with the flight anyway.
For evaluating the huge swath of conditions in between those two extremes consider your familiarity with the area (typical weather patterns, topography, obstacles, etc.) and the conditions of flight (especially day-vs-night). Thomas P. Turner's site has a nice set of guidelines which I happen to broadly agree with, but each flight needs to be carefully evaluated to determine if you believe you can complete it safely under the conditions reported to you by the briefer.
The safest approach is obviously to cancel the flight (or complete it under instrument flight rules) if you have any doubt.