Should I explain to the graduate admission committee why I am older than most prospective PhD students? Firstly, I know that there are plenty of questions here like this one but hopefully this is not a duplicate.

I was born and grown up in Iran and belong to a religious minority (Bahai faith). As you know Bahais are not allowed to attend universities in Iran because of their faith. I was no exception and so could not study at university because of my religious beliefs. Between the age of 18 (when I finished my highschool in Iran) and 24, I worked as a construction laborer. At the age of 24, I together with my family (my parents and sister) travelled to Turkey and became refugees in UNHCR  and two years later UNHCR sent us to Australia. I studied English for two years and after that I repeated year 11 and 12 because Australian universities did not accept my Iranian qualifications. At age 30, I started my undergraduate studies in Bachelor of Electrical Engineering and I will graduate very soon. I am now 35 years of age and would like to apply to be a PhD student in the first 5 or 10 top engineering schools in the world. I have maintained a GPA of 6.9, a WAM of 93 and was on Dean's merit list every year except the first year of my undergraduate studies. There is a possibility that I also get university medal but that is not certain yet.

Should I explain these details to the graduate admission committee explaining why I started my undergraduate studies very late at the age of 30? and what I was doing before that?
Such information would be relevant. The real trick would be to keep such a paragraph short and to the point. As such the question would be a good draft of such a section. I have two comments:

I would not start the second sentence with "as you know". If a committee member did not know, you make her or him feel ignorant. It is good to avoid invoking negative emotions in such a letter. Especially since leaving that part out does not change the meaning of the sentence.

It was not clear from the text whether it was the Bahai faith that prohibited it's members from going to university or whether it was the Iranian government that prohibited people with the Bahai religion to enter university.