Why does the 802.11b DSSS signal have 22 MHz of channel width? How does the information get spread over that width?
The original 802.11 standard used an 11 chip Barker code modulation.  For each data symbol, an 11 chip sequence is transmitted.  The symbol rate is 1 Ms/sec, so the chip rate is 11 Million chips per second.  This yields the resulting spectrum (i.e. null to null) is +/- 11 MHz around the center frequency.  The exact shape of the spectrum is determined by the filtering of the modulated chips.

IEEE 802.11b introduced a new modulation format called Complementary Code Keying, but it employs the same 11 Million chips per second as the original 802.11.  So the output spectrum has the same nulls at +/- 11 MHz relative to the center carrier.

See IEEE 802.11-2012 standard or 802.11 Handbook: A Designer's Companion by Al Petrick and Bob O'Hara.