Okay, Let me start with a warning first, this sentence you have doubt about describes sex , more precisely kind of a rape. So pardon me if you find the answer offensive or a few words obscene.
It is quite okay to use the word beauty to describe a man. In fact Narcissus (from whose name the word ' Narcissistic ' is derived ) was referred to as a man of great beauty. So it is common in Greek mythology and in general, olden times to use the word "beauty" to describe a really handsome man.
To answer your next question. The sentence refers to 3 people in total
1) King Aethlius - father of Endymion (who is not a boy, but a man)
2) Endymion - son of King Aethlius
3) The moon - who was allured by the beauty of Endymion
You should be clear of the three people first.
Getting to the sentence. Let me break it up
"Their first king, Aethlius, was father of that Endymion" - means, Aethlius was the first king and the father of Endymion
"that Endymion whose beauty so allured the moon" - Endymion was very beautiful. The moon was "allured" meaning, powerfully attracted to Endymion's beauty. It is worthy to note, that the moon was only attracted to Endymion's beauty, not his character.
"that she closed his eyes in a perpetual sleep" - she (the moon) was so attracted to Endymion ,that the moon (referred to as "she" in the sentence) closed Endymion's (referred to as "his" ) eyes in a "perpetual" (never ending) sleep.
"sinned at leisure" - in olden times, and even sometimes now, having sex with a person without his consent or having sex with another person who is not your spouse or having sex with people who you should not have sex with (incest) was considered a sin (trust me, it is still a sin). So they referred to immoral sex as "sinning" (Remember angelina jolie's Original sin?). leisure means free time, or you can also mean, whenever you are free and relaxed. So "sinned at leisure" means, she had non-consensual sex with Endymion whenever she felt like it. Remember, Endymion is in an eternal sleep.The poor guy cannot protest.
"and had by him half a hundred daughters." - And had because of him 50 daughters.
Finally to help you further, I will just rephrase the sentence into simple terms
Their first king, Aethlius, was father of that Endymion whose beauty so powerfully attracted the moon that she closed Endymion's eyes into a perpetual/eternal sleep and had non-consensual sex with Endymion whenever she felt like it making with him about 50 daughters in the process.