How to Brew
The fermentation of malt sugars into beer is a complicated biochemical process. It is more than just the conversion of sugar to alcohol, which can be regarded as the primary activity. Total fermentation is better defined as three phases, the Adaptation or Lagtime phase, the Primary or Attenuative phase and a Secondary or Conditioning phase. The yeast do not end Phase 2 before beginning Phase 3, the processes occur in parallel, but the conditioning processes occur more slowly.
As the majority of simple sugars are consumed, more and more of the yeast will transition to eating the larger, more complex sugars and early yeast by-products. This is why beer (and wine) improves with age to a degree, as long as they are on the yeast. Beer that has been filtered or pasteurized will not benefit from aging.
...8.2.2 Primary or Attenuative Phase
The primary or attenuative phase is marked by a time of vigorous fermentation when the gravity of the beer drops by 2/3-3/4 of the original gravity (OG). The majority of the attenuation occurs during the primary phase, and can last anywhere from 2-6 days for ales, or 4-10 days for lagers, depending on conditions.
Many canned kits will advise bottling the beer after one week or after the krausen has subsided. This is not a good idea because the beer has not yet gone through the Conditioning phase. At this time the beer would taste a bit rough around the edges (e.g. yeasty flavors, buttery tones, green apple flavors) but these off-flavors will disappear after a few weeks of conditioning.
The reactions that take place during the conditioning phase are primarily a function of the yeast. The vigorous primary stage is over, the majority of the wort sugars have been converted to alcohol, and a lot of the yeast cells are going dormant - but some are still active.
The Secondary Phase allows for the slow reduction of the remaining fermentables. The yeast have eaten most all of the easily fermentable sugars and now start to turn their attention elsewhere. The yeast start to work on the heavier sugars like maltotriose. Also, the yeast clean up some of the byproducts they produced during the fast-paced primary phase.
Leaving an ale beer in the primary fermentor for a total of 2-3 weeks (instead of just the one week most canned kits recommend), will provide time for the conditioning reactions and improve the beer. This extra time will also let more sediment settle out before bottling, resulting in a clearer beer and easier pouring. And, three weeks in the primary fermentor is usually not enough time for off-flavors to occur.