In the past, it referred to the requirement that the office-holder actually kiss the hands of the Sovereign as a symbol of personal 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 expected physically to kiss the hands of the Sovereign before assuming the role. Simply being received by the Queen is taken to validate the selection, with this meeting being described in the Court Circular as "kissing hands". The invitation issued to a party leader to form a government is sometimes still described as "an invitation to kiss hands". The actual kissing of hands does not take place until the subsequent meeting of the Privy Council, when the new minister is formally appointed as a member of the Council.
When appointing a Secretary of State (top rank in the UK Government), the protocol also involves the delivery by the Sovereign of the seals of office into the hands of the appointee. This is also valid for other officers who are keepers of seals, such as the Lord Privy Seal or the Lord Chancellor, who is also keeper of the Great Seal of the United Kingdom.