About this passagePsalm 34 is a song of thanksgiving, headed in the Hebrew text with a note tying it to an episode in David's life when he escaped danger by feigning madness. The psalm moves from public praise into instruction, and these three verses sit at the turn: they do not claim that the righteous are spared affliction, but that afflictions are many and that God is close to those breaking under them. "Broken heart" and "crushed spirit" translate physical metaphors — something shattered, something pressed flat.
The righteous cry, and Yahweh hears, and delivers them out of all their troubles.
Yahweh is near to those who have a broken heart, and saves those who have a crushed spirit.
Many are the afflictions of the righteous, but Yahweh delivers him out of them all.
In plainer words
When people who are trying to live well cry out, God hears them and gets them through it. God is close to people whose hearts are broken and who have been crushed. Trouble will come, a lot of it — and it will not be the end of them.
A restatement in modern English, written for this collection. The passage above is the text; this is one way of putting it.