The MMR vaccine essentially comes in 2 parts - the first injection at 14 months or so, the second at around 4-6 years of age. It is thought that the first injection alone is around 95% effective - in other words,only 1/20 children, when exposed to the virus, would contract it. Once the child has had the second injection, the level of protection offered raises to around 98-99% - so now only 1-2 kids in every 100 would risk contracting measles when exposed to the virus.
So the vaccine doesnt necessarily reduce the symptoms of measles ( although it might - not many studies on this particular issue tbh) - it massively reduces the risk of the child contracting measles at all in the first place.
Measles is highly contagious, so anyone exposed is very likely to contract the virus if they are unimmunized.