Why are countries near equator hot during day and cold during night?

It could depend on the winds and the moisture in the air. Also to take into account, the equator is the closest latitude to the sun in the daytime in any equinox and almost as close during a solstice, but not so close at night. Every moment the sun is heating the equator at longitude somewhere, on the opposite longitude is the equator clear on the other side of the globe away from the sun. Much as the sun heats the equator in the day, the land and air are more dry and less clouds form. Deserts are thus very common around the equator. With less cloud cover, the heat escapes the atmosphere more easily. Those are the likely contributing factors in the temperature highs and lows at the equator. The equator's two coldest hours are probably around 2:00 to 4:00 AM. I'm quite sure it is owed to the atmosphere (winds and air moisture) that heat lingers, preventing the equator from freezing up at those hours. The Tropics of Cancer and Capricorn probably do not have nights as cold as the equator, theoretically. But also, bear in mind that rainforests have a better equilibrium and that deserts generally have more extreme day heat and colder nights too. The tropic pattern can be subverted by its deserts, and the equator's pattern can be subverted by its rainforests.