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Most of the driving force behind JWST is studying the earliest galaxies. The most distant galaxies are so far, that we observe them as they were a few hundred million years after the big bang. A galaxies distance is related to it's redshift, which is the shifting of light to longer wavelengths, caused by the continued expansion of the universe. So when looking at early galaxies you need to observe longer wavelengths to see the same features we observe locally. On the other hand you might just keep your observing band fixed and deal with the fact that you can only see high redshift galaxies in the ultraviolet. The problem is that at early times the hydrogen between galaxies was neutral (not ionised) and so it totally absorbs light below a wavelength in the ultraviolet (122 nm). This means that above a certain redshift galaxies completely disappear from the visible.
My image is an example of a very high redshift quasar. On the top are images at different wavelengths, on the bottom is the spectrum. You see that below 1 micron in wavelength the quasar spectrum just cuts off, caused by hydrogen in the intergalactic medium. The quasar is only visible in the longer wavelength infrared images.
The goals of JWST are not exactly precise. It will study very early galaxies to try to better understand the origin and evolution of galaxies. Galaxy evolution is horrendously complex, it will not be solved with just one telescope.
There are other reasons for observing in the infrared too. It's less affected by dust, so can penetrate dusty regions like the centre of the galaxy, or star forming regions, or protoplanetary disks. Also material has to be very hot to radiate in the visible, the infrared allows much cooler material to be detected, such as protoplanetary disks and brown dwarfs.
Long wavelengths also offer up opportunities for studying fully-formed exoplanets, especially their atmospheres.
