It is more than just religious words. The ironborn go through a ritual when they come of age where they drown themselves, only to be resuscitated afterwards.
It was a bleak, cold morning, and the sea was as leaden as the sky. The first three men had offered their lives to the Drowned God fearlessly, but the fourth was weak in faith and began to struggle as his lungs cried out for air. Standing waist-deep in the surf, Aeron seized the naked boy by the shoulders and pushed his head back down as he tried to snatch a breath. "Have courage," he said. "We came from the sea, and to the sea we must return. Open your mouth and drink deep of god's blessing. Fill your lungs with water, that you may die and be reborn. It does no good to fight."
Either the boy could not hear him with his head beneath the waves, or else his faith had utterly deserted him. He began to kick and thrash so wildly that Aeron had to call for help. Four of his drowned men waded out to seize the wretch and hold him underwater. "Lord God who drowned for us," the priest prayed, in a voice as deep as the sea, "let Emmond your servant be reborn from the sea, as you were. Bless him with salt, bless him with stone, bless him with steel."
Finally, it was done. No more air was bubbling from his mouth, and all the strength had gone out of his limbs. Facedown in the shallow sea floated Emmond, pale and cold and peaceful.
(...)
He growled a brusque command, and his drowned men seized the dead boy by his arms and legs to carry him above the tideline. The priest followed, naked but for a sealskin clout that covered his private parts. Goosefleshed and dripping, he splashed back onto land, across cold wet sand and sea-scoured pebbles. One of his drowned men handed him a robe of heavy roughspun dyed in mottled greens and blues and greys, the colors of the sea and the Drowned God. Aeron donned the robe and pulled his hair free. Black and wet, that hair; no blade had touched it since the sea had raised him up. It draped his shoulders like a ragged, ropy cloak, and fell down past his waist. Aeron wove strands of seaweed through it, and through his tangled, uncut beard.
His drowned men formed a circle around the dead boy, praying. Norjen worked his arms whilst Rus knelt astride him, pumping on his chest, but all moved aside for Aeron. He pried apart the boy's cold lips with his fingers and gave Emmond the kiss of life, and again, and again, until the sea came gushing from his mouth. The boy began to cough and spit, and his eyes blinked open, full of fear.
Another one returned. It was a sign of the Drowned God's favor, men said. Every other priest lost a man from time to time, even Tarle the Thrice-Drowned, who had once been thought so holy that he was picked to crown a king. But never Aeron Greyjoy. He was the Damphair, who had seen the god's own watery halls and returned to tell of it. "Rise," he told the sputtering boy as he slapped him on his naked back. "You have drowned and been returned to us. What is dead can never die."
The Prophet, a Feast for Crows
The ironborn are a folk that almost lives on the sea. As such, it is important to not be afraid of drowning. This ritual basically does that. They drown themselves, so that they won't fear drowning again.
What is dead may never die
Basically, if you already died once (drowned), you can't die again. Them surviving the drowning makes them harder and stronger. Or at least, that is their belief.
So the addendum
but rises again, harder, stronger.
is simply part of the same motto.
