It really won't let us see just a few moments after the Big Bang, unless you consider 100 million years after the BB to be just a few moments after. That's about 13.7 billion years ago, with the whole universe being only 13.8 billion years old. So it'll let us see a little bit further back, but not all of the way to the beginning.
Currently, the furthest back in time we can see is with the Hubble telescope, which sees as far back as 13.2 billion years ago, or about half a billion years less further than JWST. Actually, the Planck telescope, which measures the cosmic microwave background radiation can see even further back than either of them, as it is looking at light that left just 300,000 years after the Big Bang, not several hundred million years after. But the Planck telescope is only measuring overall whole-sky fluctuations in light brightness from that period of time, and it's not concentrating on any particular spot in the sky, like the Hubble or JWST.
The reason that any of these telescopes can see that far back is due to the fact that it takes light billions of years to travel to Earth from these far away places. So we're seeing objects as they looked billions of years ago, not as they may look like now.
Now the reason that the JWST can see further than Hubble, and Planck can see further than either of them, is due to wavelengths of light that they are tuned to look at. Hubble was tuned to look at light going from the large-wavelength near-infrared to the short-wavelength near-ultraviolet portions of the spectrum; so that spectrum had the human-visible portion of the spectrum also included in between. JWST is tuned to look at only the full-infrared spectrum, from far-infrared to near-infrared, which means it sees more of the infrared spectrum than Hubble does. Planck on the other hand is tuned to view the microwave spectrum, which is even larger wavelength than the infrared.
The larger the wavelength the farther in distance they can see, and the further back in time they can see. Due to the expansion of the universe, wavelengths of light that were once visible by the human eyes, have gotten stretched to wavelengths that are longer and outside of human vision ranges. The further back in time and distance the light is, the longer it has been stretched. Since the Hubble could only see back to the near-infrared, its vision stops around 13.2 billion years ago, as beyond that, the wavelengths of light are way longer even than Hubble can see. JWST is tuned to see back to 13.7 billion years ago, as it can see longer wavelengths of infrared. 13.7 billion years ago is when it is expected that the first stars were lighting up. But the furthest light back after the Big Bang is now in the microwave region, and that's where Planck sees.