Why do British sitcoms have much fewer episodes than American ones?

From BBC America - Five Bizarre Things About British Television:

• The Six Episode Season
  
  Most British drama or comedy shows have a very short season. The classic amount for comedies is just six episodes per series – The Office being a prime example – and there’s one very good reason for this.

  American comedy is a producer’s medium, in which an idea is worked up, characters developed and early scripts written, and then the show is handed over to a larger group of writers to flesh out into actual scripts.

  British comedy is a writer’s medium. The scripts are almost always written and developed by one or two people, then taken to production. Graham Linehan, the writer of Father Ted and The IT Crowd, even directs his own scripts, which is a LOT of work.

From TIME Magazine - What the U.S. Can Learn from British TV:

There are underlying economic reasons for the two different models.

The goal of almost any American commercial series is to air at least 100 episodes, because that’s when it can be sold into syndication and aired as reruns on other stations or networks. That’s where the real money is.

But the BBC is publicly financed, so not every show has to earn its investment back with a multiseason run.

From TV Tropes - British Brevity:

British shows usually have a fairly small creative team.

It's not uncommon for one person to single-handedly write every episode of a show, as Steven Moffat did with Coupling, or David Renwick with Jonathan Creek.

There's more pressure to succeed, and less of a chance to make a lasting impression or develop long plot arcs. Ruin two episodes and that's a third of a season down the tubes. Some American shows that start off weak can grow their beard when the show would have long been over in the UK.

On the other hand, British shows tend to have the entire series filmed before broadcast, so shows are rarely canceled mid-season, or affected by events like a writers' strike.

Another thing to keep in mind is that the US has 114 million TV households, whereas the UK has 26 million. Fewer viewers means there is less money to be made. And less money means fewer episodes.

To expand on Donald's answer, larger budgets in the US mean many more writers. American sitcoms are typically written in large teams (20+ writers), with one writer penning an episode that includes contributions from all on the team. (Incidentally The Simpsons' writers have stated on DVD commentaries that the credited writer only ends up with about 30% of the lines in the final episode.)

In Britain however, that set-up simply does not exist. Sitcoms are typically written by just one or two writers in isolation, and they often direct too. For example, The Office was written and directed by Ricky Gervais and Stephen Merchant while The IT Crowd was written solely by Graham Linehan (a prolific comedy writer).

Subjectively, us Brits like to think of it as quality over quantity ;)

The most likely explanation is simple economics. Since the US population is approximately five times that of Great Britain, the television viewership is significantly larger, allowing for larger budgets per episode and more total episodes.

A lot of this depends on the TV schedule, and if the season was filmed before it ran on TV. Some networks will purchase a season in it's completed form. This covers the costs of production and film crew. It's much more expensive to hire a crew for 12 weeks, then for 12 days. So you could film 12 episodes quickly and air them on TV over a weekly schedule to recover costs.

Some shows like Friends become so popular the network can afford a full-time production crew. So they switch from pre-filming seasons to producing shows weekly. Keeping a crew on hand to produce each weekly show. Some shows can have operating costs over $1 million per episode.

Stargate for example ran for 10 years, and resulted in a full-time crew with their own production studio just for filming Stargate. Where as the first season of Terra Nova was all filmed at once, and after it ran it was cancelled.

There are a lot of popular British shows that have full-time schedules, but most of what you were watching were less known or popular. As a result, they were likely pre-film in smaller batches.

I think that Oliver_C has the best answer with information to back it up, but I would like to point out that there are many long running British shows too. Are You Being Served, Last of the Summer Wine to name two. The Office wasn't a hit when it first aired. It could be described as "brilliant but cancelled". Coupling aired in the U.S. and was quickly canceled. One really good thing about a writer's medium show is that the series can have an actual end instead of a sudden cancellation. If only some U.S. shows could be written like that. Heroes would have been a good series. Firefly would have had a proper ending within the series. But overall most British shows such as No Heroics are written not to become a hit but to be well liked.

Another aspect that I didn't see mentioned is the fact that a lot (not all) of British shows are publicly financed (BBC/Channel 4). While a lot of the American shows are privately financed (network TV).

As a business, if you have something that's making you money, you want to leverage that as long as you can.

Whereas as a service for the public, one can focus on the quality of the content over making a dollar.

That said, I do see American television slowly adopting the British model. Recent examples would include Fargo (10 episodes), the return of the X-Files (6 episodes), and Married (10 and 13 episode series). On average, longer than 6, for sure, but significantly shorter than the typical 20+ we're used to here.

UPDATE:

In updating this answer based on PhistucK's excellent points int he comments, I also came across this article:

5 reasons why TV networks are ordering shorter seasons http://www.businessinsider.com/why-tv-networks-are-ordering-shorter-seasons-2015-6

Somewhat click-bait, but I do think some of the reasons are valid points:

1. Star Power
2. Storytelling can be more dramatic and focused
3. The syndication model has changed
4. Year-round programming
5. Shorter seasons help to prevent fatigue