use and BEGIN are executed as soon as they are compiled. Without the BEGIN, the following would happen:

• Spiderman's use strict; is compiled.
• Spiderman's use strict; is executed.
• Spiderman's use warnings; is compiled.
• Spiderman's use warnings; is executed.
• What was in Spiderman's BEGIN is compiled.
• Spiderman's use Doctor::Octopus; is compiled.
• Spiderman's use Doctor::Octopus; is executed.
• require Doctor::Octopus is executed.
  • Octopus's use strict; is compiled.
  • Octopus's use strict; is executed.
  • Octopus's use warnings; is compiled.
  • Octopus's use warnings; is executed.
  • What was in Octopus's BEGIN is compiled.
  • Octopus's use Spiderman; is compiled.
  • Octopus's use Spiderman; is executed.
  • Spiderman is already loaded, so it's not loaded again.
  • Spiderman is already loaded, so it's not executed again.
  • Spiderman->import doesn't exist, so it isn't called.
    This is the problem. Exporter hasn't been loaded yet, and the variables telling Exporter what Spiderman should export haven't been initialized.
  • The rest of Octopus is compiled.
  • What was in Octopus's BEGIN is executed.
  • The rest of Octopus is executed.
• The rest of Spiderman is compiled.
• What was in Spiderman's BEGIN is executed.
• The rest of Spiderman is executed.
• Doctor::Octopus->import is called.

Remember that

use Module
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means

BEGIN {
   require Module;
   Module->import if Module->can('import');
}
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