The universe could be trillions upon trillions of years old, such that we can never see the bulk of it because it is beyond our visual horizon.

However, if that was the case, then there would be nothing at the observable edge of the universe. It would just be darkness. But that's not what we see. When we look to the observable edge of the universe, we see a glow coming from every direction. That glow, despite being very dim and very redshifted, is our glimpse of the Last Scattering event when the plasma that birthed the universe finally broke apart into separate atoms and began to emit photons.

Since stars emit specific wavelength signatures, we can see how much those signatures have shifted to determine the amount of cosmic expansion that has happened between us and any given source. We see that objects farther away are more redshifted. The Last Scattering surface appears to be 40 billion lightyears away, and using the redshift we measure, we find that the light from Last Scattering must have been emitted 13.8 billion years ago.