Why doesn't the 71% water of the earth dry or evaporate?
There are two ways this problem needs to be looked at.   The first is more astronomy than Earth science.  The Earth as an entire system is largely contained.   Its gravity and magnetic field retains nearly all of its elements.   Earth does lose hydrogen and helium and cosmic rays will split water molecules leading to a loss of an impressive amount of hydrogen and as an indirect result a loss of water, but this is loss irrelevant compared to the size of the oceans.    More detail here.  Space dust, comets and asteroids contain water so some water is returned from space too.

By the upper estimate in one article, 50,000 tons of hydrogen per year works out to about 450,000 tons of water lost every year. (and 400,000 tons of oxygen added as a result).    Compared to the mass of Earth's oceans those numbers are small.  450,000 tons per year, or 450 trillion tons over a billion years is nothing compared to the 1.3 million trillion tons of water Earth has in its oceans.   By the highest estimate, it will take 30 billion years at the current rate and at the Sun's current luminosity for Earth to lose just 1% of its oceans.  (Will look to update with other estimates).

As for the rest of the question, once we recognize that loss into space is insignificant, then virtually all water is continuously cycled though the water cycle or hydrological cycle.   Very little water gets destroyed or chemically transformed.   Nearly all of it, even of millions or billions of years, evaporates, or, turns into ice, or gets absorbed by plants, or seeps underground, but it always returns.   Evaporated water returns to Earth as rain.   Water that gets frozen on the ice caps eventually melts back into the oceans.   Water absorbed by plants or that seeps underground does eventually get returned to the surface by plate tectonics or volcanism.   Plants that store water return it when the plant is eaten.  Water is very hard to destroy, so it stays remarkably constant on Earth over time.