The idea of using bail bondsmen is unique to the United States and one other country: The Philippines. Other countries use police to handle finding people who skip out on bail, and that bail must be paid to the court, not to an individual at a private company. As a matter of fact, what the US does in allowing bail bondsmen to operate is considered illegal throughout the rest of the world. Bondsmen work by posting a person's bail for a fee that is nonrefundable. Many countries see it as a type of obstruction of justice, but the US does not interpret it that way.
In other countries, the idea that a person who cannot afford to make bail can pay a person a fee for freedom is unheard of, and unacceptable. The decision as to whether someone goes free when they cannot afford their own bail comes down to whether they can pay the bondsman's fee, and whether the bondsman will take the case. Even within the United States, Oregon, Kentucky, Wisconsin, and Illinois have abolished the practice. They either release people on their own recognizance (without requiring any money), or they require them to pay a deposit to the court.
There is a concern with bail bonds, in that they may be considered to be discriminatory against people who do not have much money. It can also be said to be overtaking decisions that should be part of the justice system, and failing to protect the safety of the public. Overall, the goal is simply to make sure that people who have "bonded out" actually show up for their trial date. It's based on the common law that came from England, whereby people pledged property or land to provide assurances that they would not try to leave the area before their trial came up.
The early 1800's saw the start of actual bail bondsmen companies, where a fee was paid for freedom and the company agreed to hunt the people down if they did not appear for their court date. In other countries, people who fail to appear are hunted by the police. The idea that even defendants who are found innocent are still required to pay a fee to a company outside the justice system if they want their freedom is often considered troublesome, but many believe it is better than remaining in jail.
The system, however, is highly effective at making sure people come to court, and it does not cost the taxpayers anything at all. Those are important reasons to keep it around, in the eyes of many people, and a large part of the reason the US still uses it today.