Actually, it is very easy to explain. The first and the main function of a mother is feeding, to be exact breastfeeding. So, a baby in its first days of life associates a mother with her breastmilk and a very comforting process of eating. While we eat (or suck the breast milk) we tend to smack and generate m-sounds. It is somehow physiological, I suppose. In the smacking sounds, the basic natural consonant is m-sound.
Very soon, babies start to imitate the smacking m-sounds and smacking and sucking lips movements to signal that they are hungry. But if a baby is hungry and its mother doesn't come (or the baby is too impatient) he starts to cry. If you try to combine the m-sound with crying (which is based on the a-sound) you'll get the ma-syllable. So, it very easy for a baby associate the ma-sound with a mode to summon its mother ('mama' is someone who comes when I am hungry or when I am crying because I am wet, feel pain etc). Women, as well, somehow subconsciously use this ma-sound to identify and name themselves in front of their babies.
Lately, ma-based words could be transformed into something else but mostly in all language, it remains as it is. For example, in my mother tongue, which is Russian, it remains "мама" = mama. In English, different versions (mom, mum, mommy, mama, mamma, mummy, ma, mam) always contain ma-syllable. In other European languages, the same thing... Even the Biblical word 'manna' based on this sound because manna is something just given gratis as the mother's milk. @Aki gave the excellent example "manma" = food...our first food is always the breast milk given by our mothers.
In a nutshell, the word 'mama' is an example of onomatopoeia and that is why children so easily embrace it.