How come dental pain numbing shots to cause me to feel high?

Epinephrine. Many anesthetic solutions contain epinephrine, also known as adrenaline. This can cause a temporary rapid heartbeat. It is used to prolong the effect of the anesthetic by constricting the blood flow in the area of the injection. There are anesthetics available that do not have this. Ask your dentist to use one of these if you are sensitive to the epinephrine.
High=Fast Heartrate? Some people get an increase in their heart rate from either the adrenaline in the anesthetic or to the rush of adrenaline that their body produces in response to fear, which is sometimes called the "fight or flight" reflex set off by the brain. Discuss this with your dentist so he can decide on which anesthetics to use and how to help you be calm for the injection.
Adrenaline rush. The anesthetic has Epinephrine causing an adrenaline rush. Ask the dentist to use anesthesia with no epinephrine.
Multiple possible... There are several possibilities. It could be Epinephrine in the injection or adrenaline related to anxiety with treatment. Observe other times you feel this way and consult with your dentist or physician to find similarities and diagnose the cause.
Side effect. Could be either the epinephrine or lidocaine. If the feeling is objectionable, as your dentist to try other local anesthetic options.