Why is a second bee sting more dangerous then the first bee sting to a person who is allergic to bee stings?

That's really simple: an allergic reaction is when your body responds excessively to something, making way too many antibodies to counter the antigen. When antibodies are made, Memory B cells are also made. These help identifying the antigen faster the next time.

So the next time you're stung by a bee, the Memory B cells recognise the antigen and antibodies are made faster than the first time.

There's you're answer, sorry for my English incase I misspelled something, I'm not from America.