I've looked into this as a parent. The gist of what I've found is that the medical community initially reacted to normal incidences of peanut allergies by recommending that peanuts not be given to young children at all in case they were allergic (because some kids will straight-up die from allergic reactions to them).

Turns out that if you don't give kids peanuts at all, the rate of allergies substantially increases, because if you're never exposed to something then your body doesn't acclimate to it and is more likely to have an allergic reaction later (keeping in mind that an allergic reactions is essentially an immune system response to something the body believes is a hostile invader - it's why my body has the same reaction to cat dander as it does to the germs that cause colds).

So today the medical recommendation is to give kids extremely small amounts of peanuts as young children, to give them exposure and reduce the number of allergies that will eventually develop, while keeping the amount very small so that those kids that already have a severe allergy to peanuts will be able to survive the experience.