Here's the relevant dialogue
PICARD: How old are you?
SOJEF: We came here from a solar system on the verge of self-annihilation, ...where technology had created weapons that threatened to destroy all life. A small group of us set off to find a new home, ...a home that would be isolated from the threats of other worlds. ...That was three hundred and nine years ago.
PICARD:
And you haven't aged a day since then?
SOJEF: Actually, I was a good deal older when we arrived ...in terms of my physical condition.
ANIJ: There's an unusual metaphasic radiation coming from the planet's rings.
It continuously regenerates our genetic structure. You must have noticed the effects by now.
PICARD: We've just begun to. ...I suppose you're seventy-five.
ARTIM: No. I'm twelve.
TOURNEL: The metaphasic radiation won't begin to affect him until he reaches maturity.
PICARD: To most offlanders, what you have here would be more valuable than ...gold-pressed latinum. ...I'm afraid that's the reason that someone may be trying to take your world away from you.
I think the slow motion thing was just some trick they learned over the years
ANIJ: You stop reviewing what happened yesterday, ...stop planning for tomorrow. ...Let me ask you a question. ...Have you ever experienced a perfect moment in time?
PICARD: A perfect moment?
ANIJ: When time seemed to stop ...and you could almost live in that moment.
PICARD: Seeing my home planet from space for the first time.
ANIJ: Exactly. Nothing more complicated than perception. ...You explore the universe. We have discovered that a single moment in time can be a universe in itself, ...full of powerful forces. ...Most people aren't aware enough of the now to even notice.
PICARD: I wish I could spare a few centuries to learn.
ANIJ: It took us centuries to learn that it doesn't have to take centuries to learn it.
But I expect sooner or later the planet would naturally lose the ability to sustain them.
Are there any takers for this Cellular Regeneration and Entertainment Chamber being the key to immorality? From DS9 "In the Cards"