I think the idea is that felons would inevitably vote for those who would make it easier for them to go unpunished or be permissible with their crimes, so to allow them to continue their enfranchisement would be to undermine the republic. Considering we currently have MILLIONS of felons in our country, this isn't entirely unfounded... but at the same time, it also prevents systemic review of the justice system by preventing those who undergo it's abuses to have any control over how it s run. This has let our justice system be turned into a self-serving engine to funnel prisoners into for-profit prisons, which often have little oversight in how they are run.
Basically, it's a sticky issue. If we allowed them to vote, it's a sure bet that any organized crime factions would vote in lockstep to undermine our ability to fight them. If don't, then we lose the insight of those closest to how the system runs. Organized crime has historically been a HUGE problem in the US, so we choose to fight it instead.