The Cincinnati Enquirer reports that Governor Kasich vetoed a provision of a bill that would require out-of-state college students to obtain an Ohio driver's license 30 days after registering to vote in the state. This aspect of the bill was criticized by many Democrats as a "poll tax," as it would have attached a cost to registering to vote in the state for certain potential voters. According to the Enquirer, this provision was part of a highway spending bill that establishes rules for when new permanent residents of Ohio must obtain Ohio licenses and register their vehicles with the state.
It provided that registering to vote serves as the intent to make Ohio your permanent residence and that then triggers the requirement to obtain an Ohio license and plates within 30 days. Governor Kasich vetoed this part of the bill, leaving the 30 day requirement in place for those who are moving to Ohio permanently. Determining if a move to Ohio is permanent is based on whether there is "a specific intention to return to live in a different state," per the Ohio Administrative Code. According to the Enquirer's report, the Ohio BMV will consider registering to vote in the state as evidence that you intend to make it your permanent home, but it is not a declaration of permanent residency that requires you to change your license and registration.
As explanation for his veto, Governor Kasich asserted that existing laws were already successfully determining residency and cited the possibility of creating confusion for both Ohioans and government institutions. See the full text of the bill and his statements, here.