Know that we have divided
In three our kingdom; and ’tis our fast intent
To shake all cares and business from our age,
Conferring them on younger strengths, while we
Unburden’d crawl toward death.
The old king is looking to divide up his kingdom. To this end his daughters engage in a brown-nosing contest to try to determine who loves him best. Cordelia puts her foot in her mouth clear up to her knee and thus is left out in the cold.
The Earl of Kent doth protest too much about this turn of events. He almost ends up impaled on the king’s sword but cooler heads prevail. He is sent packing, with orders not to return under pain of….you know what. Cordelia’s stock has now plummeted but the king of France decides to have her as his wife even so.
In other developments, there's some sort of duplicitous machination type stuff involving Edmund and Edgar, the sons of the Earl of Gloucester. I haven't quite sorted this out yet but perhaps it will become clear as things unfold.