From the French word that gave us ‘to gallop’. In the sixteenth century it was a word for a boy working in a kitchen, a turnspit (OED) but occurs much earlier as a by-name.
Final ‘th’ was regularly not pronounced in some dialects, notably in a suffix such as ‘forth’, so ‘garr’ was a characteristic regional spelling of ‘garth’.
In general usage a 'gate' is a form of barrier, a door, set across a gap in a wall or hedge, and it can be opened to allow people and animals to pass from one place to another.
In numerous northern land deeds the word ‘gate’ referred to a right for animals such as cattle, horses, sheep and swine to ‘go’ onto certain grazing lands, it was a right to pasturage.