We can see a planet 1000 light years away in that detail?
We cannot. The background of that image is not the planet or the star (and is rather confusing).
JWST resolution limit is about 1.9E-7 radians at 1000nm wavelength. At 1000ly WASP-96 (the star) would subtend about 7E-11 radians so is hopelessly unresolvable. WASP-96b is much smaller. Indeed if orrbital radius from Wikipidia (0.045au) is correct, is not possible for JWST to resolve planet as separate from star even (this is about 7E-10 radians).
This is not what they are doing however: WASP-99b moves across its star as seen by us, so what we do is watch the star as it does so: the light from the star is (slightly) blocked by the planet, and some of the light also passes through the atmosphere of the planet so the spectrum of the light that reaches us changes slightly. From these changes in the spectrum we can conclude things about the atmosphere of the planet. Particularly significant is evidence for clouds which WASP-96b was formerly thought not to have.