The term "valediction" means "farewell". Thus this is a "goodbye" poem in which a lover is leaving his beloved. The departure can represent both a temporary one of the separation of living lovers but also represent the ultimate form of departure which is death.
Similarly to Donne's other poem on the same theme, "Sweetest love, I do not go", this poem suggests that true lovers should not mourn either temporary separation or the death of the beloved. The reason for "forbidding mourning" upon separation is religious.
The Reverend John Donne was a priest in the Church of England. That means that he was a dualist who believed that mind and soul were independent of the body and that the soul persisted after death. From a religious viewpoint, there are two types of love, a carnal sublunary kind (focused on "eyes, lips, and hands") and a spiritual kind. Spiritual love rather than being thwarted by absence of a lover's physical self becomes stronger, because in the absence of the physical, what remains is "so much refined" as to be purely spiritual.