There is no proof in the sense that we know this for an absolute fact, but the following is the best approximation we can give, which is based on strong evidence but there are a few possible problems. You might want to brush up on your knowledge of special relativity, specifically how the relativistic Doppler effect works, as it is necessary to understand how we know how old the universe is.

The entire sky has a glow called the cosmic microwave background (CMB), which is the afterglow of the big bang: it is the light from the primordial plasma that the universe came out of as it expanded from the big bang. The temperature of this plasma when we are able to see it is the minimum temperature by which a plasma becomes opaque, i.e. light cannot pass through it, which is about 3000K (if light could pass through it, we could see all of the way to the big bang itself). However, because the plasma is moving away from us so fast, it has been red-shifted by the relativistic Doppler effect to have a temperature of about 3K, which is why we see it as a microwave background in the sky instead of a visible light background. By measuring the apparent temperature of the CMB, we can calculate that it has been red-shifted by roughly 1000x. You can then use Einstein's relativistic equations to calculate how fast it is moving away from us (it's very close to the speed of light).

The next thing you need to know is how fast the universe is expanding. We do this through what are called "standard candles" - events that have a certain known brightness that we can use to determine how far away they are based on how bright they appear to us in the sky. One of these are type-Ia supernovae, which are what happens when a white dwarf gets overloaded by mass from a companion star and goes supernova. This has a finite mass because it's the threshold at which point white dwarf stars are no longer possible (it's about 1.44 times the mass of the sun). Using these, we calculate that the rate at which the universe is expanding is about 70 kilometers/second per million parsecs. This number is called the Hubble constant. So, for example, a galaxy located ten million parsecs away from us will be moving (on average) about 700 kilometers per second away from us.

By dividing the speed that the CMB is moving away from us by the Hubble constant, we can calculate how far away the CMB is. The reason why the CMB is the distance that it is is because light has had only a finite time to travel. Thus, by dividing this distance by the speed of light, we calculate the age of the universe. It so happens that the distance to the CMB is about 4.3 billion parsecs, or 13.7 billion light years. Thus, the universe is roughly 13.7 billion years old.

All of the above is calculated based on known physics, but there are a few places that we don't know the exact details, which is why I objected to your use of the word "proof" in your question. These are:

1. We don't know exactly how bright Type-Ia supernovae are. If they are brighter, then they are further away, so the CMB is further away and the universe is older. If they are fainter, then the universe is younger.
2. The rate at which the universe is expanding is changing with time. We would have to integrate the rate of change in the expansion rate of the universe over the universe's age in order to get a "true" value. We only approximately know how the rate of expansion is changing.
3. We can really only measure the age of the primordial plasma itself. Our theories imply that the universe was very young when the primordial plasma became traversable to light, but if some kind of exotic physics was taking place back then (and it may have been), then it is possible that our estimate of the age of the universe could be wrong because of it.
4. Related to #3 above, general relativity predicts that there was a singularity at the beginning of the universe: a point of zero volume and infinite density, in which time literally began. The Heisenberg uncertainty principle from quantum mechanics states that this is impossible. Obviously, one of the two theories must break down at this point as they cannot both be right, but we have no experimental verification of which of these otherwise well-tested theories that might be.