Our word "gymnasium" is taken almost directly from the Greek. A Greek gymnasium was a public school almost, a wide yard with various exercise grounds, instructors and weights in different forms over the years. Plato was himself a respected wrestler and has several quotes where he states that it is a waste to not develop your body. Galen, the famous Roman physician of the coliseum, is another source for a lot of past exercise information -- apparently a popular exercise for Roman athletes was to wear heavy lead weighted mittens and to swing your arms around.

The connection between lifting heavy things and gaining muscle definition and strength is taken as granted in ancient texts. Aristotle also mentions dieting in his discussion of the mean, stating that while a person can eat between one and nine pounds of food Milo (the paragon of ancient athletes) should eat six -- that the mean is good but the goodness of the mean is relative to other factors and not just the bare average of the numbers.

I don't remember what the quote was from our about (it was likely an ancient taking about how things worked in Sparta) but I remember reading that it either was or should be codified in an ideal city that men not productively married (so, with kids) by a certain age should be forced to spend their days near the gymnasium. The idea was the image of all the beautiful young exercising bodies would provoke the bachelor into a more civic arrangement.

There is one ancient greek fable about a wrestler who in order to gain strength, carries a baby ox around a track each day. As the wrestler becomes stronger the ox also grows, making for a harder challenge.

Are you referring to weightlifting/exercise for the purpose of increased physical attractiveness alone, or some other trend of aesthetic advantage to having a heavy object held up in the air of which I am unaware?

I'd like to point out that I'm not a historian, but I'm a physical education teacher that had to go through history and science of exercise classes to get my degree.

Everyone here is mentioning the Greeks. Exercise arose all over the globe in different forms at different times and for different purposes from health to appearance to military readiness.

While not a detailed source, the powerlifter Jamie Lewis has a series of articles that he wrote earlier this year about the history of exercise and various implements which can be found here: Part 1, 2 and 3. The articles are easy to read, and Jamie cites the majority of his sources. I encourage you to read his articles as well as look into his source lists as well if you want to learn more.

Dumbells existed in Ancient Greece. In fact, the word for dumbbells in my native language, portuguese, is halteres, the same greek word.

Could anyone give insight in other parts of the world? Specifically interested in East and Southeast Asia.

Are there any records that talk about how the greeks discovered that lifting heavy things would make them stronger?

No one has sufficiently answered the question so far. The OP is asking about weight training for the sole purpose of aesthetics. It's easy to prove that historical peoples from many cultures and time periods used weight training to increase power, and also that these people understood the concept of modifying food intake to bulk up or cut down size, but is there any proof that any pre-20th century people weight lifted for aesthetics and no other reason? Almost always there's another goal, such as military preparation, training for sports, exercising for health, etc. These people weight lifted with strength as the primary goal, so that they would be more effective at some other activity. Improved aesthetics was just a by-product of training. Ideally what we need is a source that describes someone exercising with the singular goal of becoming more attractive, and not to be a better soldier, fencer, wrestler, etc.