What is required for a pilot to fly in another country?
For your specific example, Philippines Civil Aviation Regulations Part 2 (Personnel Licensing), section 2.2.4 covers "Validation and Conversion of Foreign Licenses and Ratings." It states that a holder of a license from a "Contracting State in accordance with ICAO Annex 1" (this includes the United States) is eligible for "validation" of their foreign license. The validation process involves presenting the appropriate documentation of your license, ratings and logbook. For more than just PPL privileges, you must also pass a "skill test" and demonstrate the required knowledge and ability. The conversion ("validation certificate") gives you the same ratings you had on your US license and is valid for 1 year.

The FAA has a very similar policy for foreign-to-US license conversion, covered in 14 CFR §61.75. As in the Philippines, the FAA does not require a skills test for private pilot privileges, but a written knowledge test is required for conversion of an instrument rating.

Another question here on ASE discusses conversion from EASA (Europe) to FAA (US).

Most other ICAO states recognize each other's licenses in a similar manner. In all cases you have to present your license and proof of experience, and in some cases pass a written test and/or checkride. I haven't look up the exact regulations for, say, EASA (Europe) or CAAC (China) but I'm sure they aren't far off from the examples above. In reality diplomatic considerations probably also come into play: a holder of an Iranian license would probably run into problems attempting to convert/validate their license in Israel, even though both are ICAO-contracting states, since they aren't on very good political terms to say the least. But this will likely be the exception rather than the rule.

(There are only 3 non-ICAO states: Dominica, Liechtenstein and Tuvalu.)