How is the sea water purified into drinking water in United Arab Emirates?

The process is called desalinisation. There are a few processes, but the main ones is done by heat.

It involves taking seawater (which is water containing dissolved salts). Heating the seawater to boiling point and allowing, the pure water evaporates as steam, the steam is collected and condensed back into fresh water. The dissolved salts cannot evaporate become concentrated as "brine", this concentrated salt water is then pumped back into the sea, and the fresh water pumped to consumers.

Most emirates use natural gas (which is abundant in the region) to heat the water to boiling point.