Why do Muslims Fast during Ramadan

All Muslims, Shia and Sunni, fast during the month of Ramadan (Ramazan), i.e. they eat something, called Sahari, before Azan of morning prayer (about 1:30 hour to sunshine) and then don't eat anything during daylight hours until Azan Maghrib (about 15 minutes after sunset). They also must not drink, have intimate relations, swim, etc. After Azan Maghrib, they break their fast (Iftar).

Now, the question is Why do Muslims Fast during Ramadan?

Simple answer is that it is Wajib(obligatory) in Islam for healthy adults according to Quran(2:183):

O you who have believed, decreed upon you is fasting as it was decreed upon those before you that you may become righteous -

However, it has several advantages which some of them are described in Hadith of Miraj in where the prophet Muhammad(pbuh) ascended to the heaven and talked with Allah:

The Prophet Muhammad (pbuh) asked Allah: O God! What is the start of worship? He said: start of worship is fasting and silence. My Lord! What is the effect of fasting? Said: fasting leads to wisdom and wisdom leads to knowledge and knowledge leads to certainty; and when a person comes to the position of certainty, it’s no matter, his life is easy or difficult
Part of Hadith Meraj, BiharAlanvar, V 74, P 27

Arabic and Farsi translations

رسول الله صلی الله علیه و آله قال: یا ربِّ ما أوّلُ الْعِبادةِ؟ قال أوّلُ الْعِبادة الصّمْتُ و الصّوْمُ قال یا ربِّ و ما مِیراثُ الصّوْمِ قال الصّوْمُ یُورِثُ الحِکْمة و الحِکْمةُ تُورِثُ الْمعْرِفة و الْمعْرِفةُ تُورِثُ الْیقِین فإِذا اسْتیْقن الْعبْدُ لا یُبالِی کیْف أصْبح بِعُسْر أمْ بِیُسْر
من حدیث المعراج، بحارالانوار ج 74، ص 27

رسول الله صلی الله علیه و آله:
خداوندا! اول عبادت چیست؟ فرمود: آغاز عبادت، روزه و سکوت است. عرض کرد: پروردگارا! اثر روزه چیست؟ فرمود: روزه باعث پدید آمدن حکمت است و حکمت موجب شناخت و شناخت موجب یقین است، پس هرگاه بنده به مقام یقین رسید، در بند سختی یا آسانی زندگی نیست
بخشی از حدیث معراج، بحار الأنوار (ط - بیروت)، ج ‏۷۴، ص: ۲۷

Ref: AndishehOnline