Tricky question. I'll use personal examples. Here goes: "at the age of" describes an event that happened once at a particular point. "At the age of 9, my parents got divorced." "by the age of" describes something that took time to happen, like building a skill. This is usually used to show advanced children. "By the age of 5, I could already read and write extremely well." "the age of" actually puts the specific age as the subject of the sentence. "The age of 14 was tough for me because I was very shy around girls." "with age" sounds like it belongs in an adage. A piece of wisdom that talks about growing older and its benefits. "With age comes understanding."

'At the age of' means, for example, when someone is that age. eg I learnt to play the piano at the age of five. 'By the age of' means at or before that age. eg Most children have learnt to speak by the age of three. This means that some maybe learn to speak at the age of one or two, but by the time they reach three, most children have already learnt to speak. 'the age of' on its own often refers to a period of time. eg The nineteenth century was the age of steam 'with age' means as a person gets older. eg Most people's hearing deteriorates with age. I hope that's all clear.