Precession is a real astronomical phenomenon: the Earth's axis traces a slow circle every ~26,000 years. This shifts the position of the equinoxes against the constellations.
How fast?
About 1 degree every 72 years. Slow enough that no individual notices, but fast enough to add up to a full sign over 2,160 years.
The Age of Aquarius
The famous ages (Pisces, Aquarius) refer to which constellation is rising at the spring equinox. We are slowly leaving the Age of Pisces.
Does it change my sign?
Only if you switch zodiacs. See tropical vs sidereal. In tropical astrology, your sign does not change.