I recently read a Reader's Digest article about how this phrase almost didn't make it in the speech. The Pentagon and State Department wanted it out, but Reagan himself left it in "because it was the right thing to do".
Peter went out among the people to get a sense of their sentiment and learned they had strong feelings about the wall. I heard he asked at a dinner party and was met with silence. Then, when the people heard he was sincere, they opened up the floodgates of their feelings about that monstrosity.
Peter penned the famous phrase and everyone down the line tried desperately to have it removed from the speech. The State Department was apoplectic! Reagan asked Peter to explain why he wrote it that way, understood the meaning, agreed with it completely, and overrode everyone. He kept in the phrase and delivered the address. Important concepts communicated simply and clearly.
I'm sure there is a better explanation than mine in Robinson's book, "How Ronald Reagan Changed My Life"