Engagement rings have a long history. They have been used since at least Ancient Egypt. The circle-shape of the ring was supposed to symbolize a never-ending cycle of love. They were in use during Roman times too, until they fell out of use until they picked up again in the 13th century.
One of the traditions that many people follow in the United States is that the groom is supposed to spend 1 month's worth of salary on an engagement ring. This tradition was actually started by diamond company De Beers in order to sell more diamonds. Go figure. They later suggested that a man should spend 2 months' salary, in an effort to sell even more diamonds.
This had an unintended consequence, though. Before 1935, men who reneged on an engagement were liable to being sued for breach of promise. The damages tended to be costs incurred in preparing the wedding, emotional distress, and loss of other marriage prospects.
Starting in 1935, laws that permitted men to be sued for breaking up an engagement started to get repealed or limited. The costs of breaking off a marriage, though, stayed the same. They were particularly bad if the woman was not a virgin, as it would be more difficult to find another partner. As a result, those expensive diamond rings became a type of financial insurance for the woman in case she needed to sell it for the money.