Howdy and welcome to Nikonites!
The key piece of information that you're looking for is the "AF-S" in the lens description. Where they are multiple different 105mm f/2.8 prime models that Nikon has built, the AF-S has the built-in autofocus servo so that you can use it completely with the D5300. You might also find older AF-D and earlier variants of this lens. Those earlier models expected the autofocus motor to be on the camera body, which unfortunately the D5300 doesn't have.
As Marcel pointed out, as long as the lens doesn't list DX on the barrel, it'll work on both DX and FX bodies. Then, as long as it's an AF-S lens as well, you'll also be able to autofocus on your current body.
Hope that helps.