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We see only stars whose light is not older than the age of universe. This is understood. If a star is too far away that currently we don't see it then possibly we will see it later, then the age of the universe will grow.

What I don't understand is how a star that we see now can move away out of visible universe in future.

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Due to the finite speed of light combined with the expansion of the universe, and that the rate of that expansion increases the further away you look, this results in a far-away "particle horizon" beyond which, previously emitted light cannot reach us. The particle horizon is, however, expanding.

As time progresses though, the "future horizon" moves gradually inward due to the accelerating nature of the expansion, meaning objects we could once see will slowly disappear. That is, while light from some point emitted long ago can reach us now, light emitted now from that point might never be able to reach us.

I should note that the future horizon is currently further away than the particle horizon.

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