Why was there a government in-exile during the Japanese occupation in the Philippines?

Answer 1

Simply because the President, Manuel Quezon, was spirited away to Australia by submarine from Corregidor Island off Bataan.

Even as the Battle of Bataan was getting more and more desperate and as the humidity and staleness in the air at the tunnels of Corregidor worsened Quezon's TB, Douglas Macarthur thought it prudent to have him and his family spirited away to Australia while 'the getting was good.'

Answer 2

Because they escaped the Japanese cruelty.