Why do mosquito bites become large red lumps under the skin after you scratch them?

Histamine. Scratching the skin releases histamine the bodies itch and swelling chemical. Its release leads to redness and swelling.
Histamine. Usually it's histamine released from blood vessels, dermis get swelling.

Related Questions

My daughter had what I thought at first to be a simple mosquito bite that has turned into a big red spot on her inner thigh with a lump under the skin?

Bite. It can be a large local reaction to the bite, especially if she is scratching it. Ice it and put some Hydrocortisone cream on it. If it does not subside, and if it gets worse, let her doctor look to see if it is infected.
Infection/abscess. Some of the insect bites will become infected with bacteria and form an abscess beneath the skin. You can make a simple warm compress with a moist wrag in a zip loc bag, microwaved until warm and applied to the skin. This can increase the circulation in the area and mature the site. It could either go away or come to a head and drain. By tomorrow, you can decide if a visit to the doc is needed.
Insect Bite...!!! Dr. J. Ferguson advised warm compress, dr. N. Munmeh- cold; confusing? Any insect bite clean with soap water, if itchy, bumpy: cold compress 1st few hours, hydrocortisone; can give oral benadryl: to reduce allergic response. If after 1-2 days getting even bigger, deeper: warm compress; if not better in couple of days- see her pediatrician to rule out possible infection and treatment if needed.
Not an insect-bite. It is more likely your daughter has furuncle (so called boil). Staph skin infection can initially present as small blistery lesion similar to insect-bite. Have her checked.