Pence never actually said that Kaine “ran a state that failed,” although he did say he comes from a “state that works.” He offered some explanations for that claim, but let's set those aside for now and look at the most obvious: people working.
It is true that Pence's jobs record as governor of Indiana (running from January 2013 to the present) is better than Kaine's (January 2006 to January 2010). In Virginia, the unemployment rate rose and the number of people employed fell during Kaine's time in the executive mansion. The opposite was true for Pence.
You've probably already figured out the big HOWEVER that accompanies this. But let's walk through it anyway. Here's how unemployment evolved in each state under each governor.
Notice that the unemployment rate in Virginia spiked about two-thirds of the way through Kaine's administration. That was right about when the country had a big recession, as you may recall.
Comparing the state figures to the national numbers, it's clear that neither Virginia nor Indiana was particularly exceptional. Virginia's unemployment rate rose as the national numbers did; likewise, Indiana's fell.
That drop in Indiana was steeper than the national drop, except for a blip earlier this year. As a relative change from the unemployment rate when each man took office, the unemployment rate rose a bit more in Virginia than it did nationally and it fell a bit more in Indiana than it did nationally.
If we do the same analysis with the number of people working in each state, the picture flips. In Kaine's Virginia, people lost jobs at a slower rate than nationally; In Pence's Indiana, people gained work at a slower rate.
That Kaine governed at the outset of the recession and Pence during the recovery is critical to Pence's overall point. He said Kaine left office with a deficit (which isn't really accurate, given the need for a balanced budget in the state) and bragged about having $2 billion in the bank, about the same amount he inherited when he came into office. He lambasted Kaine for trying to raise taxes, a proposal Kaine made toward the end of his term out of the requirement to keep the state's budget balanced.
All of that said, the argument that Pence's tenure has been better on jobs than Kaine's is true. Pence's Indiana went back to work. Kaine's Virginia lost jobs. The easy lesson: Don't hold executive office during a brutal economic downturn.