What is a teleological argument:
A teleological argument explains a thing or event (phenomena) by the purpose or function it serves as its reason for being. This is opposed to explaining phenomena by the elements and events that cause it. Teleology is an "end justifies the means" kind of argument. For example, plants produce oxygen so that animals can breathe; likewise, animals produce carbon dioxide so that plants can use it in photosynthesis.
Using a teleological argument to support the existence of God, the philosopher would justify the existence of God based on God's function in the world and to humanity. Some teleological arguments for the existence of God could be based on statements such as: there is clearly a design in the world. It is based on such complex balance and symbiosis, that only an omniscient being, such as God, could have put it into place. Everything has a purpose and its form and structure seem designed to fulfill that purpose. Therefore, such a complex design and integration of things could not have happened accidentally. An intelligent being, God, must have designed such a world/universe; God must have designed things to have a design.
Aristotle called this an explanation based on "final causes." In other words, Aristotle thought that some things exist because they are striving to realize their true nature or their true purpose. A tomato grows from the ground because its purpose is to do so. Clearly, there are root (pun intended) causes that lead to the seed being planted, adding water, soil and sunlight that make the tomato plant grow. But there is also the teleological argument that the tomato's purpose/function (the very definition and event of its growing and ripening) that is also a cause in its development and existence. It exists because it is designed to develop. This would be like saying something exists because it is designed to reach its final cause, its final purpose. Therefore, God exists to fulfill his purpose which is to design all things with a purpose, including himself. In this case, God is his own cause.
How does the teleological argument fail to prove the existence of God?
The problem with the teleological argument is that it ignores root causes. If God exists to fulfill his own purpose, which is to design all things, then he would have to have designed himself. Therefore, using the teleological argument there is nothing prior to God, no cause that led to God's existence. With no explanation for God other than the final cause, the final purpose, it is difficult to say how God manifested in the first place. Even with the tomato plant, it certainly has a final cause (the ripe fruit), but it was clearly caused by seeds, soil, water, and sunlight. The teleological argument for the existence of God gives no physical evidence of how God came to be. The teleological argument only proposes that God exists because God's purpose is/was to design all things, including himself. In other words, God was designed to exist (designed by himself?) and therefore he exists. The teleological argument fails because there is nothing/nobody to design God. For the teleological argument to succeed, the philosopher would have to explain how God got his design/purpose/final cause in the first place. It's difficult to explain how God could have designed himself to be the designer of all things (including himself). Following the teleology, wouldn't God need to be designed to design himself to be the designer?