Because mosquito's teeth is sharp. No you are wrong.
Actually never even bite. It sucks our blood through straw like structure called Proboscis. The proboscis are made of 6 parts ,out of which 4 parts are used to pierce into our skin and blood vessels, and hold the tissues apart .Through the 5th part called Hypopharynx the mosquitoes drop it's saliva which contains anticoagulant. Anticoagulant prevent the blood from clotting so that the 6th part of Proboscis called Labrum suck up the blood.
Now our immune system recognises the anticoagulant as an invader and it releases Histamine. This histamine dilates the blood vessels So that the blood along with immune cell come to attack the invader and heel the effected area. This Histamine causes the itchy feeling.