Where does "patching through" come from?

The original telephone switchboards were literally patch boards: the caller and callee were physically connected with a wire (patch cable) to form a circuit.

Photo from here.

Macmillan Dictionary has an entry for patch through:

patch someone through to allow someone speaking to you on a phone or radio to speak to someone else
I’m just patching you through now.

Dictionary.com's 18th definition of patch also has that meaning.

The etymology is directly related to the terms @Useless commented (and then answered) about: patching through used to mean connecting the two calls with a patch cable (often called a patch cord or patchcord) on a patchboard (which is often called a patch panel). The phrase still applies, though, even if there isn't a physical patch panel.

The etymology of the usage of patch in this context is probably from using temporary connections, as opposed to permanently wired ones.

To take it a little further back, they were called patch boards, I believe, because they were used for temporary connections in some electrical systems, so they were a patch to make something work for a while, before, sometimes, the system was hard wired.