Woz explains why the original Apple II (1977) doesn't support lowercase letters. That's not surprising, in retrospect; neither of its major contemporaries, the Commodore PET 2001 nor the TRS-80 Model I, does either, despite being developed by major corporations with substantial resources.
He doesn't explain why the Apple II+ (1979)—after the II's market success was proven—doesn't support lowercase letters. Even if software uses graphics mode to display lowercase letters, the II and II+'s keyboard does not have physical/electrical support for detecting shifted letters. Since graphics mode is cumbersome and slow, word processors for the II and II+ typically use reverse video to indicate capital letters, and use another key like Escape as a shift toggle. A popular alternative is the shift-key mod that fattire mentioned, which requires soldering of a wire to one of the paddle ports.
The lack of support is because the company was working on the Apple III (1980), which it expected would quickly obsolete the II series. The III has built-in 80-column text and full lowercase support, at both the character-font and physical-keyboard levels. Apple had incentive to not make the II too attractive.
Neither Woz nor anyone else at Apple expected that a) the III would quickly fail, and b) the II series would remain Apple's bread and butter. Without the III's distraction the II+ would surely have had built-in lowercase software and hardware support, or there would have been another II around 1981 with such. As it were, the III took up so much of Apple's resources that the Apple IIe did not appear until 1983, by which time the IBM PC had surpassed the II series.