"The speech lady says he will grow out of it." The speech therapist is saying that the child is probably developmentally delayed and will eventually speak in a manner that other people will understand.
"My brother was backward in speaking." There is a good chance that Mum's brother was also developmentally delayed. As he matured (moved forward in age) his speech improved.
"Out it all comes" means that thoughts and words that have been repressed (unspoken) for a period of time often are verbalized when individuals become angry or confrontational.