What is happening here
A responsorial psalm is the assembly's sung response to the First Reading at Mass, drawn from the Book of Psalms. A cantor or psalmist sings the verses, and the people answer with a recurring refrain called the antiphon. It falls between the First and Second Readings, and it is a prayer rather than a second reading: the congregation's reply, in the words of Scripture, to what they have just heard. These are the very prayers Christ prayed in his earthly life and that the Church prays without ceasing in the Liturgy of the Hours.
The structure is simple. A cantor or psalmist sings the verses; the assembly sings the antiphon (the recurring refrain) in response. The antiphon is usually drawn from the psalm itself or from the day's Gospel, and it captures in a single line the spiritual heart of what the assembly is being asked to make its own.
The Responsorial Psalm is properly sung, not recited. The General Instruction of the Roman Missal is direct on this point: "the responsorial psalm should as a rule be sung." The reason is that the psalms themselves are songs; they were written to be sung, prayed in temple worship, prayed by Jesus, prayed by every monastic community since the early Church. To recite a psalm is possible but always second-best.
A few examples show how the antiphon distills a whole psalm into one line. For funerals, Psalm 23 is the near-universal choice, with the refrain "The Lord is my shepherd; there is nothing I shall want." For weddings, couples often pick Psalm 128, "Blessed are those who fear the Lord and walk in his ways," or Psalm 34, "Taste and see the goodness of the Lord." On an ordinary Sunday the psalm and its antiphon are set by the lectionary and change week to week, always chosen to echo the First Reading. The psalm a couple or family chooses shapes the sound and feel of the whole liturgy.