The process of making vaccines can be quite complicated. The most simple explanation for the involvement of eggs in the FluMist live attenuated vaccine is that they are used to allow vaccine virus replication. Eggs are also used in the production of inactivated vaccines as well.
For Flumist production, scientists take the chosen predicted strains of flu for the upcoming season and complete several processes. The virus is adapted and attenuated in order to produce a virus capable of mounting an immune response (protection against that virus) but not capable of causing influenza-like illness. Special eggs that are free of all pathogens (infectious particles) are inoculated with the attenuated virus so that the virus can replicate. After this, special filtering is done to remove the virus from the egg and produce the vaccine.
If you are allergic to eggs, you should talk with your doctor or pharmacist about receiving a different flu vaccine such as Flublok. Always discuss the risks and benefits with your healthcare provider, especially when an allergy is involved.