Why do insect bites make you itchy?

In the case of mosquitoes their bite has with it a substance that numbs the area being bitten (actually, mosquitoes don't bite, they stick). It lasts just long enough for the mosquito to do its thing. Then the numbing quality wears off and it becomes an irritant.

it would very much depend on the insect but just like when you cut yourself or graze yourself even when your body is healing it becomes mildly irritable, alot of the time its the same with insect bites.

When your skin is inflamed and the bite is a little swollen then it is not caused by your bodies natural healing process

when they bite they inject some acidic poisonous substance

histamine that is why you put an anti-histamine (aka anti-itch cream) like benadryl. this pill or cream also helps seasonal allergies (the cream doesn't help the seasonal allergies just the pill).