In the past, it referred to the requirement that the office-holder actually kiss the hands of the Sovereign as a symbol of their fealty and loyalty to the Sovereign, that fealty and loyalty being a requirement to serve in the King's or Queen's government. In modern times, office holders are not required to actually physically kiss the hands of the sovereign (though some, to the current Queen Elizabeth II's embarrassment have unexpectedly done so). Simply meeting with the Queen is taken to validate the appointment, with this meeting being described as "kissing hands".
The invitation issued to a party leader to form a government is sometimes still described as "an invitation to Kiss Hands".