Well I think it's all in the title..anyway..I would like to know through which passages did Binet get to his matrix multiplication algorithm and where (book, paper..) did he first tell about it to the world
You might want to peruse some findings of of Oliver Knill (Harvard):
You'll find a number of potential resources to dig further. It is mentioned that Binet discussed such an algorithm in a lecture dated November 30, 1812.
From Wikipedia: Binet is "recognized as the first to describe the rule for multiplying matrices in 1812."
If it's of any solace, you might like the quote you can find on the linked webpage: