A "tape out" used to be labeled as "tape monitor - out" on most receivers, integrated amps, or preamps, and eventually was shortened to just "tape out" most of the time.
Back in the days when most folks had a reel-to-reel or cassette deck attached to their analog systems, it served quite a useful purpose. The "tape out" basically just replicates the signal from the selected source. Back in analog days, that was usually the output from the phono preamp section, so it could be sent to the input of the tape deck to record your LP's onto tape.
The other set of jacks on the amp could be labeled "tape in", "tape monitor", or "tape monitor - in". They were connected to the signal coming back from the tape deck's playback head, which was located just after the recording head in the tape path. That way, you could listen to what was recorded on the tape only a fraction of a second after it was recorded.
That served a couple of needs......first, you could hear if you had set the recording level too high on the tape machine and ended up with a distorted recording, and secondly, you could hear if the tape had a "dropout", a defect that prevented the signal from recording properly. You'd hear a split second of silence, then usually your own expletives as you realized you had to re-record the whole LP on a new tape.
The "tape monitor" switch on the preamp/receiver/integrated amp allowed you to listen to either the source directly, or the signal coming back from the tape deck.
If you didn't have a tape deck, or if you had multiple tape monitor loops, you could connect an external device like a graphic equalizer in the same way. The tape monitor allowed you to easily switch back and forth to compare the unequalized vs. equalized sound.
In all cases, the signal level from the "tape out" jacks is unaffected by the setting of the volume knob on the preamp/integrated amp/receiver.
Even though analog tape is pretty much ancient history for most of us, the terminology remains.
