How does strep throat lead a child to post streptococcal glomerulonephritis?

Antibodies. The body makes antibodies to the strep which cross react with the kidneys and the body attacks it's own tissue when the immune system is tricked.
Immune response. After certain infections (such as strep throat), the body makes proteins called antibodies to protect us from the infection. The antibody binds to the proteins from the infecting agent (antigens) and make an immune complex (a larger protein) which can deposit in the tiny glomeruli (functional units of the kidney) and start an inflammatory process locally with resultant damage and glomerulonephritis.

Related Questions

Explain why glomerulonephritis is a common result of strep throat in children?

Occasional not comon. Certain strains of the untreated strep germ can trigger formation of cross reactive antibodies. Their primary target would be the strep germ, but for unknown reasons they attack some of the kidney tissue as if it were strep. Damage to the tissue produces glomerulonephritis.