The medieval Catholic focus on the dead at the time of All Hallows Eve is at the root of Halloween as we know it.. By the fourteenth century a custom called 'souling' had developed in England in which the poor would go from house to house asking for soul-cakes. The better-off would give out small cakes or loaves in exchange for prayers for their dead relatives. Souling continued up until the twentieth century in some parts of Britain, though the ritual became increasingly secularised and was eventually relegated to children. Souling almost certainly forms the basis for American 'Trick or Treating'. Shakespeare uses the phrase 'to speak pulling like a beggar at Hallowmass'. I'll note that both of my parents, who grew up in Detroit, Michigan in the 40s and 50s, refer to the practice of trick-or-treating as 'begging' and to trick-or- treaters as 'beggars'. The phrase they used to ask for treats as children was 'Help the poor!'

It began in Ireland in the late 1800's, when children, on All Hallows Eve, would sing songs to the dead in exchange for cakes from willing adults. It was known as "souling", and is still practised in some parts of Ireland and Scotland.

Each cake the child collects is symbolic of a soul freed from purgatory.

It comes from the older practice of guising and souling in Scotland/Ireland and England.

Halloween being the night when the walls between the living and the dead worlds were thin and souls could walk the earth.

Souling is the practice of saying a prayer for the dead in return for a small present,

Guising is rooted in the Celtic purification rituals - Guisers would go round dressed up as evil spirits and perform some small performance (originally to pacify spirits later just a party piece) in return for a gift.

Emmigrants took this to the US (The Colonies) where over time it changed.

In the UK this tradition to some extent got tangled up in Guy Fawkes night on the 5th Nov with it's accompanying "Penny for the Guy" and Bonfires (although fire was a big part of the older pagen festivals) and is being gradually weakened by the American version of demanding gifts for NOT doing a trick.

From the wikipedia article on trick or treating:

The practice of dressing up in costumes and begging door to door for treats on holidays dates back to the Middle Ages and includes Christmas wassailing. Trick-or-treating resembles the late medieval practice of souling, when poor folk would go door to door on Hallowmas (November 1), receiving food in return for prayers for the dead on All Souls Day (November 2). It originated in Ireland and Britain.[3]

Trick-or-treating is the Halloween tradition where children go door-to-door dressed in costumes and saying "trick-or-treat" as a request for candy. It is said that the tradition began in the Middle Ages and mimics medieval "souling", a day when the poor went door to door receiving food in exchange for prayers for the dead. It is said to have originated in Britain and Ireland.