They were one of the core components of the Books of Hours, along with the Penitential Psalms and Office of Our Lady, and the way the books are organised suggest that most lay people could be assumed to know them by heart.
But the reasons for their popularity go back, I think, to the Fathers' enthusiastic endorsement of them. Accordingly, today I want to provide some Patristic texts that help explain why this occurred for us to meditate on.
St Augustine and Jacob's Ladder
St Augustine, for example, explains their significance as follows:
"For it is, according to the title prefixed to it, A song of degrees. Degrees are either of ascent or of descent. But degrees, as they are used in this Psalm, are of ascending...There are therefore both those who ascend and those who descend on that ladder. Who are they that ascend? They who progress towards the understanding of things spiritual. Who are they that descend? They who, although, as far as men may, they enjoy the comprehension of things spiritual: nevertheless, descend unto the infants, to say to them such things as they can receive, so that, after being nourished with milk, they may become fitted and strong enough to take spiritual meat...When therefore a man has commenced thus to order his ascent; to speak more plainly, when a Christian has begun to think of spiritual amendment, he begins to suffer the tongues of adversaries. Whoever has not yet suffered from them, has not yet made progress; whoever suffers them not, does not even endeavour to improve. Does he wish to know what we mean? Let him at the same time experience what is reported of us. Let him begin to improve, let him begin to wish to ascend, to wish to despise earthly, fragile, temporal objects, to hold worldly happiness for nothing, to think of God alone, not to rejoice in gain, not to pine at losses, to wish even to sell all his substance, and distribute it among the poor, and to follow Christ; let us see how he suffers the tongues of detractors and of constant opponents, and— a still greater peril— of pretended counsellors, who lead him astray from salvation..."
Early Christian writers found much to reflect on in the symbolism of the number of psalms. Cassiodorus, for example, explains that:
"Their purpose is to unfold in fifteen ordered steps the blessedness of the faithful people, celebrated in that previous song in which their wide-ranging merits are assembled, to elucidate the mystery of the New and Old Testaments. The number seven, as has often been said, denotes the week occasioned by the sabbath of the Old Testament; the number eight signifies the Lord's day, on which He clearly rose again, and this is relevant to the New. When joined together, they are seen to make up the number fifteen. The psalmist begins with renunciation of the world, for he shudders at the worldly ways which constitute the burden of his ills. From this base he mounts by the steps, so to say, of merits, and reaches the perfect and eternal love of the Lord, which as we know is set at the very summit of the virtues.
We explain this progression more clearly in its due place. But I think that I should advise you that through the bounty of divine grace, fifteen steps are laid in these psalms to denote in various ways the saints' merits, just as there was the same number in the temple at Jerusalem, which we know was completed by Solomon. This was so that the present order of the psalms, prefigured in that building, should be seen to be foretold, for that earthly construction seemed to bear the likeness of the heavenly temple. So when we hear the word steps in the psalms, we are not to think of anything material to be mounted by physical movement, but we should interpret it as the mind's ascent.
The word canticle has been placed first so that we may apply it rather to the progress of the soul. Step here is the ascent of humility, confession of sins, as was stated in Psalm 83: In his heart he has disposed to ascend by steps, in the vale of tears. We shall deserve to mount these steps only if we prostrate ourselves for our sins. So let us continually entreat the Lord....
Some commentators think that the fifteen additional years accorded to king Ezechias are related to this parallel, so that the number fifteen is shown to have signified also the course of his perfect life. The prophet speaks throughout the psalm. In the first part, he cries to the Lord, asking to be delivered from wicked lips and a deceitful tongue. In the second, he is fiercely afflicted because in lingering too long in this life and enduring the vices of others he is weighed down by an assortment of ills…"