When you hear "state" for laws old enough to be common-law, think more of "community." A murderer makes random strangers in town mad on the victim's behalf, so they go kill the murderer. They aren't killing them to get justice for the victim; they're killing them because the victim was someone they knew as a member of the community and they feel pain at their loss. This pain, in aggregate, is the pain the "state" feels in response to an act, and lashes out in response to. It's the reason that terrorism is a bigger deal than plain treason: both affect the government itself in some sense, but terrorism inflicts pain on the people in aggregate, and so the people-in-aggregate, in their roles as "legislature" and "jury", seek to punish it more strongly.

If that were true, murdering someone with little to no connections (e.g. a homeless person) wouldn't be a crime.

As a citizen my motivation for punishing murder is to keep society stable and my family safe. It doesn't matter who was murdered, it matters that the perpetrator is tried.