Question about public transportation in Tokyo

I have this hyperdia app for my smartphone and i looked up how i can travel from Shibuya to my hotel (at Asakusabashi St.).
You can look up the results here: http://i42.tinypic.com/142df75.png
My question is:
How can this only cost 210 Yen? That's very cheap in comparison to the public transportation in Germany or England. And how come the fare is only for the first ride of the several needed? Is the ticket for the yamanote line also valid for the JR Chuo? Thank you so much in advance

How can this only cost 210 Yen?

Because the entire trip can be done using a single railway company. And because a lot of stereotypes that Westerners have about Japan are wrong or out-of-date.

And how come the fare is only for the first ride of the several needed?

Because the fare is calculated for the entire trip and not by adding up the cost for separate section fares. It happens that Hyperdia displays it that way.

Is the ticket for the yamanote line also valid for the JR Chuo?

Yes, it is the same company. So you can use a single ticket. Or use Suica at the ticket gates in Shibuya and Asakusabashi.

To give you another example, on the Tokyo Metro fares index there are the following sample fares:
Fares depend on the distance you will travel.
Distance Fare
1km~6km 160 yen (Child: 80 yen)
7km~11km 190 yen (Child: 100 yen)
12km~19km 230 yen (Child: 120 yen)
20km~27km 270 yen (Child: 140 yen)
28km~40km 300 yen (Child: 150 yen)

The trip you checked for JR trains was 17.1 km..long so the fare you got is in a similar price range.

The better route is:

Shibuya (JR Yamanote line) to Yoyogi (JR Sobu line) to Asakusabashi

You just change your train once at Yoyogi. It may take about or less than 30 minutes. The fare is 210 yen for a single ticket with no doubt.