end of the growing season

What does the "end of the growing season" mean? I think this just applies to summer crops, because if you are growing fall/winter crops they have to continue growing past the first frost date of the gardener's particular area.
Am I correct?

For most folks growing vegetables it means winter or whenever the ground freezes. In your area, you may harvest a bit of kale in January, but that is about the only vegetable that will stand a hard freeze while producing. If you don't grow kale or something similar the season ends with the first killing frost.

'End of season' means whatever you want it to mean.
If you can grow things over the winter, then the 'end of the (winter) season' is just in time to clear a few beds for the warm season crops.

However, in most places in the US there is a definite winter, with snow, or at least freezing temps. 'End of the season' generally means the end of the warm season, major clean up. A small amount of cool season crops may go on into the early winter. But basically the beds are cleared, winter mulch spread, and so on.

So if my growing season ends the last week of October, when do I start cabbage, lettuce or spinach? I have to germinate the seeds out of the garden, I cannot direct seed due to the heat and humidity. When do I direct seed carrots, beets, turnips, radish?

In your area of Virginia you can direct seed turnips in August, radishes in September. Fall cabbage need to be transplanted in July. Never had much success with spinach, beets, or lettuce as a fall planting in Virginia so I can't help you there. Check with your Amherst county agent. http://offices.ext.vt.edu/amherst/

If cabbage is transplanted in July that means "fall" cabbage has to be started in May. If I put cabbage in my garden in July how does it survive the heat and humidity of July and August, even September? I tried cabbage a few years ago in September and it pretty much melted away.

Cabbage is pretty tough. I grew up a few miles.north of You in Fluvanna County. We always transplanted Late Flat Dutch or Drumhead just after July fourth for winter storage. Usually cut afetr the first killing frost, placed in a kiln covered with straw and enough dirt to be below the frost line and then removed for use when weather permitted during the winter months. In later life, I lived in Pulaski county where I actually had a root cellar.