Translation:I will teach the piano starting the week after next.
"I will teach piano the week after next" would actually be "saraishuu piano o oshiemasu" -- however, the sentence from the question was "saraishuu KARA piano o oshiemasu". The "kara" here means "after" or "from _ onwards", which is why the accepted translation was "I will teach piano starting the week after next"
"From the week after next I will teach piano" is close to a literal translation without doing violence to the English language why then do you deny it. There is no reason whatsoever to exclude "from" as a correct interpretation of から , the idea of starting may be inferred, but it is by no means explicit or indeed essential in conveying the notion that the speaker will teach (the) piano after an interval of a couple of weeks . By the same token, I could equally justify the phrase correctly as "In two weeks time I start teaching piano". It is irritating to have to guess at every turn which extra word you will throw into the mix.
The English on this app is so awkward. What is the point of just figuring out sentence order in unnatural English when the word blocks are there, you need to use all of them and the result is the weirdest English ever. I thought I was here to learn Japanese, not do English puzzles in bad grammar.