In the tips and notes, it gives the example: 'Я ви́жу, как она́ танцу́ет. = I see her dancing.' Not 'I see how she dances', which would mean something different. So why is 'они видят, как ты живёшь.' 'they see how you live'?
Sometimes words mean what they mean. If you can imagine a question "How do you cook rice?", then you may, in principle, have a sentence "I learnt how you cooked rice".
But you ARE right, this sentence does not follow what we teach in this skill. I removed it.
Don't remove good sentences just because they don't follow the rule! I just wanted to know why it meant that; if it's a bit irregular, then it's it's a bit irregular. But if people say it, then it's still worth knowing.
I still don't understand, however, how you tell when как mean 'how' and when it indicates perception of action: is it just context? How would you say 'They see that you live'? (They, perhaps, tried to kill you, but now you've escaped, and they've noticed it.) Or is 'to live' not a verb of action, but just a state? In which case, it all makes sense.
It's just the Russian style. You put commas in front of что, как, которой etc.