Why can human viruses that can't infect chickens be grown in embryonic chicken cells?
The measles vaccine uses an attenuated virus: one that has been  modified to reduce virulence such that it is (hopefully) no longer harmful to the host but still retains the antigenic determinants to induce acquired immunity. Although in modern times rational attenuation by genetic engineering is becoming established [1], the traditional method involves simply growing the virus under conditions abnormal for it such that it must evolve to survive. This often involves culturing the virus in cell types (or possibly organisms) in which it does not typically cause infection. Due to existing variation within the viral population or novel mutations, the virus may, over time, adapt to infect these cells with some degree of efficiency. Importantly in the context of vaccine development, the genetic changes generated during this adaptation may also reduce the pathogens ability to causes disease in the host organism [2].
This effect was originally observed by Louis Pasteur in 19th century who subsequently used it to develop a vaccine against the bacterium Erysipelothrix rhusiopathiae, which infects pigs [3]. Pasteur noticed that propagating this bacterium in rabbits, in which it did not initially cause disease, increased its pathogenicity in rabbits while at the same time reduced its pathogenicity in pigs. The virus had evolved and this attenuated form was used as a vaccine in pigs.
The attenuated virus for measles vaccine was created by the same principle in the 1950s to 1960s. An isolate of the measles virus from an infected patient was successively passaged through human kidney cells, human amnion cells, fertilized hen's eggs [4] and chick embryonic cells [5]. This attenuated virus, which was now replicative in chick embryonic cells, was no longer infective in humans and was used as the basis of the measles vaccine [6].

References
[1] Bull JJ. 2015. Evolutionary reversion of live viral vaccines: Can genetic engineering subdue it? Virus Evol 1(1):vev005
[2] Badget MR, Auer A, Carmichael LE, Parrish CR, Bull JJ. 2002. Evolutionary Dynamics of Viral Attenuation. J Virol 76(20):10524–10529
[3] Bazin H. 2003. A brief history of the prevention of infectious diseases by immunisations. Comp Immunol Microbiol Infect Dis 26(5-6):293-308
[4] Milovanovic MV, Enders JF, Mitus A. 1957. Cultivation of Measles Virus in Human Amnion Cells and in Developing Chick Embryo. Exp Biol Med  95(1):120-127
[5] Katz SL, Milovanovic MV, Enders JF. 1958. Propagation of Measles Virus in Cultures of Chick Embryo Cells. Exp Biol Med 97(1):23-29
[6] Enders JF, Katz SL, Milovanovic MV, Holloway A. 1960. Studies on an Attenuated Measles-Virus Vaccine — Development and Preparation of the Vaccine: Technics for Assay of Effects of Vaccination. N Engl J Med 263:153-159