There is a generic answer and a uniquely Jewish answer to this question.
The generic approach is to look at the universe and employ some form of the argument from Design.
The argument from design works on two and a half levels:
For those who reject evolution (another lecture) the argument comes from irreducible complexity we find in the biological world. Components which make up DNA and other machinery of the cellular world work and even look like machines. Michael Behe who is a microbiologist has done extensive research on this topic and is one of the leaders of the Intelligent Design movement.
For those who accept the evolutionary explanation for the development of all biological life, there is still the huge mystery of “where did the simplest life form come from.” Even atheists are really at a loss for this question.
Yet another argument comes from what is called “The Anthropic Principle” Essentially it is the same argument from design which is altered to bypass the theory of evolution. The physical sciences can only explain HOW things in the physical universe work. They cannot be used to explain WHY things work in the first place. What is a star? They explain that a star is made up of helium and hydrogen atoms colliding, causing a nuclear reaction called fusion. That’s a description of what is happening and how stars are formed from these elements. But to explain why there is helium and hydrogen existing in the universe at all, and why they have these physical properties and produce stars when they collide—those are why questions which cannot be answered even by the most detailed description of the process itself.
Science finishes its job when it fully describes what takes place and describes the necessary physical conditions which bring out the processes we see. All the why questions can never be answered by science and you need religion for that.
Unfortunately, we erroneously feel we have explained something completely when we describe what happens and the immediate causes. Supplying a physical sounds like an adequate answer to a why question, but it isn't. That shouldn’t satisfy the really curious person who yearns to know the real ultimate causes and reasons for the very existence of the universe and everything in it.
We need to posit the existence of G-d to explain any ULTIMATE why’s of the universe. Why are the physics so fine tuned to allow the existence of biological life as we know it?
But there is a deep problem when we are positing the existence of God in order to resolve an unsolvable mystery. How meaningful is that? Do we pray to him? Does this Creator care about us? And this function of God-as-Creator is a very man-centered one. He only exists because we need Him to exist in order to solve a philosophical question.
So much for the generic answer.
The Jewish answer is that we received the answers from direct experience.
Both of the above questions have the same ultimate answer: Yetzias Mitzrayim.
The Ramban at the end of Parshas Bo says that the question of G-d’s existence and His creation of the universe is resolved from the quality of the ten makkos. Hashem attached a message to various makkos explaining what their function was and what belief they were designed to instill.
Some makkos showed Hashem complete and total ability to manipulate and co-ordinate the laws of nature. He systematically smashed every Egyptian source of power and object of worship in order to illustrate that he created the natural world. He also illustrated his awareness of the details of human events and intervened in history to save one nation from another. He is the creator and supervisor of this world.
This was going to be a one-time event
The Ramban concludes that Hashem only openly reveals Himself and proves his existence and role as creator in this manner once in human history. He will not perform miracles for every skeptic who demands direct proof. He made a big enough illustration of this truth as to make it unlikely for future generations to have any serious doubt to its occurrence and unnecessary to repeat it ever again.
