Why were people sent to concentration camps?
The entire Nazi terror apparatus, including sections of the police, had the authority to send people to concentration camps. For example, sometimes the police decided that people with previous convictions were repeat offenders and, instead of sending them for trial, simply sent them to concentration camps.
Did the prisoners in those concentration camps starve to death because the allies cut off the food supply to Germany?
No, British soldiers taken prisoners by the Gerrmans were sent to prisoner of war (POW) camps, where conditions were much better. A very small number of British prisoners - for example, some of those who kept on trying to escape - were sent to concentration camps. There were also a very small number who were sent to concentration camps by mistake.
The extermination camps (without forced labour camps) were only used for Jews and Romanies/Sinti (gypsies) only and were dismantled and in two cases grassed over when they had 'served their purpose'. Other categories were sent to ordinary concentration camps. ___ Handicapped people were killed in separate centres, and political prisoners, communists, trade unionists, habitual criminals, homosexuals, Jehovah Witnesses, beggars and vagrants were sent to ordinary concentration camps.
They did not "go" to concentration camps, they were forced into them at gunpoint. The Nazi's reason for sending them there was to exterminate them. The vast majority of people who were sent to concentration camps were murdered within hours of their arrival. The Nazis intended to kill every single Jewish person in Europe.