Large stars emit much of the there energy in the visible and near-ultraviolet wavelengths, but a galaxy at z=10 will be observed most brightly in the infra-red. Moreover, lots of the interesting spectral lines (such as Hydrogen alpha) will be shifted to the infra-red.
But there are other reasons for wanting a space infra-red telescope. Infrared is less absorbed by dust, allowing us to see through dust clouds. And several interesting objects glow more brightly in IR: brown dwarfs and large exoplanets, for example. But IR is absorbed by the atmosphere, so land based telescopes are inferior.
Now very distant galaxies are in a universe with a lot of hydrogen atoms. (in the universe now, most of the hydrogen is ionised forming an intersellar plasma) And hydrogen atoms absorb photons at wavelengths shorter than 91.2 nm (in the ultraviolet). For a galaxy at z=9, this will be redshifted to 912nm (in the near infrared). The galaxy will be very dim at optical or ultraviolet wavelengths.
So, the reason for using this band is: Distant galaxies are brighter in this band, there is lots of other science that can be done, and that science can't be done with land based telescopes.