Uncle Axel makes David promise to tell no one that he is telepathic, because if anyone finds out he will be in great danger.

Uncle Axel discovers that David has been “talking” to Rosalind, even though she is not nearby.  He talks to her in his head.  Axel becomes very concerned.  He is worried about David, and makes him promise to tell no one.

I want you to keep it secret. I want you to promise that you will never, never tell anyone else what you have just told me -- never. It's very important: later on you'll understand better how important it is. (ch 4)

Uncle Axel also insists that David not do anything that would lead someone to become suspicious.  In their society, anyone who is different is destroyed.  David’s difference is not a physical one, so he should be able to keep it hidden.

It isn’t until this conversation that David really realizes what the threat is. He is faced with this situation in trying to protect Sophie as well.