"He wouldn’t come out of the house that much. When I did see him drive by, I’d wave and he wouldn’t wave back," said Gareth Crites.
I can't sense the difference.
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Do is sometimes used for emphasis.
See definition 2 do, definition 4 a:
— used to make a statement stronger
'You really do look lovely today!'
In your example the emphasis is probably used to offset the negativity of the previous statement. "He wouldn't come out of the house that much. When I did see him drive by ..."
The speaker just finished suggesting that he doesn't come out of the house much, so the fact that he may be seen driving by should be slightly surprising, and deserves some extra emphasis.
"When I did see" is much more emphatic the "when I saw."
When we want to stress something, we can use "do" or "did" (depending on the tense we need to use) before the notional verb:
I hardly ever leave the country. When I do leave the country, I try to make the most of the trip.
He rarely spoke to his neighbors. When he did speak to them, everyone was happy.