Say, in Japan, a person commits a crime. The law only punishes the person that committed the crime, but Japanese society will usually make life hard for the family of that criminal, probably more so compared to other countries, like the UK, or the US. Most Japanese people think that the responsibility lies not only in the criminal himself, but also in the family members.
(Although not all think like this. I certainly don't, and many others don't either. But it seems to be a cultural thing that Japan holds 'responsibility' quite high up in their priorities, compared to, say, 'redemption'.)
This is the case with the Takakura family, and why Shouma holds the burden of his parents quite heavily on his shoulders.
If Shouma was a Brit, I'm pretty sure he would have acted completely differently. It would have been something along the lines of 'My parents were arseholes. Screw them.', and he would have been more ashamed than feeling responsible himself.
I think this very Japanese view of 'responsibility', is what the 'destiny' in the show is. It's the view that society forces things onto people and forces onto them a kind of 'destiny'.