What does it mean to be a product of one's time?

It means that you are in agreement with the ideas and values of the time you grew up in. For instance, if you were British in the 19th century, you would genuinely believe you had the right to an empire and that other races were inferior. If you were French in the 20th century you would expect a job for life and accept a great deal of government control. People all over the world, in most cultures, have happily accepted the concept of slavery. The idea is that generally-accepted ideas change over time, and it is not fair to judge someone who lived hundreds of years ago by today's standards. Not, of course, that today's standards are always better!