I'm trying to hookup my DVD player and stereo VCR to my Vizio HDTV.
The cable installer used composite cables between the Scientific
Atlanta set-top box and the TV when we got HD service. I'm going to
replace that connection with an HDMI cable, to free up the composite
inputs on the TV.

I'm confused about SA's instructions for the VCR. They say I need an
RF coaxial cable and 2 sets of AV (RCA) cables, if I want to archive
shows recorded on the Explorer's DVR or watch pre-recorded tapes. Why
would I need the RF cable? Wouldn't one set of RCA's to move stuff
from the Explorer to the VCR, and a second set to move stuff from the
VCR to the TV, be all I need?

I'm trying to hookup my DVD player and stereo VCR to my Vizio HDTV.
The cable installer used composite cables between the Scientific
Atlanta set-top box and the TV when we got HD service. I'm going to
replace that connection with an HDMI cable, to free up the composite
inputs on the TV.

You can not get HDTV over a composite connection. You need a component connection (three video wires) or HDMI. If you are using a composite connection you can only see HD downconverted to standard definition.

The cable installer used composite cables between the Scientific Atlanta set-top box and the TV when we got HD service.

Are you sure you mean composite (single yellow jack) and not component (three red, green and blue jacks) video? An installer who uses composite video to connect an HD box and HDTV is incompetent and you should complain to the cable company about him. As Rory noted, you can get only SD over a composite-video connection.

Why would I need the RF cable?

Probably to allow you to use the VCR's tuner to record an analog channel while you're watching a digital channel via the SA box's tuner.