When cities were enclosed by heavy walls for protection, the keeper of the key to the gate was a powerful and respected figure, usually the king (or prince, baron, mayor...). This is whence came the custom of presenting an honored guest with the "key" to a city: today it is a symbolic gesture indicating respect. By naming Christ Scepter of Israel, we announce his kingship; by naming Him the Key of David, we proclaim that He is not only the one with the power to open the gates (the key holder), but also the very opener (the key itself). Today's Antiphon immediately put me in mind of this verse:
And I will give to thee the keys of the kingdom of heaven. And whatsoever thou shalt bind upon earth, it shall be bound also in heaven: and whatsoever thou shalt loose upon earth, it shall be loosed also in heaven.
Matthew 16:19, DRV
Christ comes to open the gates of Heaven to us, and he entrusted that key to the Church through the authority of St Peter, the first pope.
O come thou Key of David, come
And open wide our heav'nly home.
Make straight the way that leads on high
That we no more may have cause to sigh.