Matter originated in the Big Bang some 15 billion years ago. How do we know this? Well, we don't "know" this as a certainty. One of the points we should make is that NOTHING is known for certain in science. That is the nature of the discipline we call science. While nothing is known for certain, a great many things are thought to be so firmly established that one would be foolish to treat them as unlikely.
Now the consequence of making a flawed judgment can vary enormously when it comes matters within the domain of science. First of all, what do we mean by this? What IS science, and what is its domain? We must define our terms with some care, especially today, when it is common to hear, for example, that evolution or global warming are "just" a theories.
The notion of the "scientific method" is rather familiar to students who have taken high-school level courses in chemistry, physics, or biology. In such courses, one learns that the scientific method consists of [1] collecting information (data), [2] formulating some hypothesis (or tentative theory), [3] testing the hypotheses, and [4] possibly modifying the hypothesis to fit new data. It isn't common to mention that this method is widely used in everyday life, but it is.
If you hear that there is a new grocery store in the Westgate Shopping Center, you may treat this as information collected--step [1]. You may consider this a possibility, a hypothesis--step[2]. You may test this hypothesis by driving to the Center to see for yourself--step [3]. After this the hypothesis may be confirmed, in which case you may consider it "true"--step [4]. But there is no absolute truth in science. The store may relocate, in which case, you will modify your hypothesis. You may choose to consider relocation a trivial or a significant modification of the original hypothesis. If the store closes, the modification is surely significant.
This all may seem trivial when put in such an everyday context. We tend to think of science in terms of academic disciplines, chemistry, physics, astronomy, or the life sciences. We don't think of the problem of how to buy groceries as a matter of science. Certainly, we don't think of primitive man's problem of finding and killing game as a scientific one. However, the methods used involved the scientific method as surely as those which have led us to date the time of the Big Bang.
It has been said that there was no science before Galileo (1564--1642). This is hyperbole, but Galileo studied a broad range of questions in physics and astronomy, and left an extensive written account of his endeavors. However brilliant and accomplished his predecessors might have been, we have limited knowledge of their efforts. We deduce that Archimedes (ca. 287--212 BCE) and Aristarchus of Samos (ca.320--250 BCE) were brilliant, but on the basis of fragmentary and anecdotal evidence.
Prior to Galileo's time, many scientific questions were submitted to "authorities" for answers. Authorities were both secular and religious, and there was considerable overlap. The Catholic theoretician St. Thomas Aquinas (1225--1274) made extensive use of the writings of Aristotle.
When one appeals to authority, it is generally assumed that the truth lies within the writings or thoughts of that authority. Apparent conflicts of new observations with that authority are to be resolved by a more careful interpretation of the sayings or writings of the authority. In other words, the "truth" is presupposed to be contained within the authority. Those who believe in the ultimate truths of sacred texts take any apparent conflict with observation as an imperfection of the interpretation of those texts, and not of the texts themselves.
This approach contrasts with the modification of a scientific hypothesis on the basis of new data. This does not mean there must be a conflict of religion and science. There are many religious scientists, some deeply so. Among astronomers and physcists, religion and science can be compatible because they deal with separate questions--those of the physical world, and those of faith. Galileo was reputed to say that the Bible tells us how to go to heaven, and not how the heavens go.
How do we know when the Big Bang took place? The simplest answer is that we observe the most distant objects in the universe to be receding from us. We can measure their distances and their velocities. If the basic relation distance equals velocity times time holds (d = Vt). We can then solve for the time given the measured distance and the velocity.
The age we get by applying this formula is called the Hubble Time. The accepted age of the universe, about 14 billion years, is somewhat less than the Hubble time, but the difference is unimportant here.
The simple formula d = Vt, gives us d = 0 at t = 0. Does this have any meaning? Obviously no measurements could be made at t = 0, or anywhere near such a "time." How far back in time can we meaningfully extrapolate? There is good reason to believe that we can make measurements now that apply to a time a few hundred thousand years after the Big Bang, or t = 0. These measurements are of the cosmic background radiation (CMB), and they are the subject of intense modern research. This time, t = several times 105 years, may seem like a rather long time, but it is only about 0.01% of the age of the universe. The current age of the Earth is rather well known from radioactive dating; it is about a third the age of the universe itself.
On the other hand, we believe that all of the hydrogen and helium, elements that dominate the chemical composition of the universe, were made in a small fraction of that time represented by the CMB. Indeed, these elements were made from more primitive materials, quarks and radiation, within the first three minutes of the time since the Big Bang.
There are no direct observations of the universe during the first several minutes of its existence. But we can make inferences of what was happening during this time with the help of theoretical models. We think we know the current density of matter and radiation in the universe now, and we know the rate of expansion. From this we can use a model to calculate the density of matter and radiation for any time in the past. We can go back nearly to t = 0, but current theories won't let us go all the way to t = 0. Some say we can use current models all the way back to t = 10-43 seconds, but to understand the composition of material coming out of the Big Bang we only need to consider the universe at an age of a few seconds to a few minutes.
Within the first several minutes, we can use rudimentary ideas, e.g. d = Vt, and physics that can be tested in the laboratory, to calculate what the chemical composition of matter in the early universe would be. It is mostly hydrogen and helium, but there are trace amounts of isotopes of these and other elements. The relative composition of these materials allow us to tune our ideas about the expanding universe with some precision.
We can observe the CMB radiation that comes to us from times when the universe was several hundred thousand years old. We can calculate the conditions in the universe when it was a few seconds to a few minutes old by considering the relative composition of the isotopes of light elements, which we can observe now. We need some formula or recipe to connect the observations to the age of the universe. A simple example is d = Vt. In general, formulae that describe physical systems allow us to model these systems, that is, calculate their properties. A great deal of astronomical research is connected with the computation of models that reproduce observed properties of planets, stars, nebulae, galaxies, and the universe.
The ages of the universe that we have discussed are really ages of a model (or of models) that simulate the properties of the observable universe.