A phoenix is a mythological bird that dies as it reaches old age (literally by burning up, or, if you want, spontaneously combusting) and is reborn (young and beautiful once again) of its own ashes. It is a creature associated with a whole slew of related concepts: long life, rebirth, resurrection, regeneration...
In the context of the sentence you quoted,
Japan was a phoenix destroyed by the war and now, 70 years later, [is] one of the most economically powerful countries in the world.
the author is saying that the country was destroyed in the war, but rebuilt itself to its previous strength (or beyond it) — quite possibly in a spectacular, unexpected way. Note that often the new phoenix is considered a different creature from its predecessor (its "parent") — similarly, post-war Japan is different (though obviously related) from pre-war and wartime Japan.