Re: Why do particles behave as wave-particle duality?

Wavelength=Planck's constant / momentum.
This, de Broglie's formula, is a fundamental law of nature, without which 
the world would be completely different and life as we know it could not 
exist - for example, it is quantum theory that is responsible for the 
periodic table, and therefore all of chemistry. Thus, the answer to your 
question is the anthropic principle - if this were not the case, you 
wouldn't exist to notice that it was not the case. You can only ask why it 
is or is not true because it is true.
Perhaps recent, as yet incomplete, 'theories of everything' such as 
superstring or twistor theory will show something more fundamental (such 
as the vibration of a string in 10 dimensions) to lie behind wave-particle 
duality, but the question "why is this so?" will always remain.
In the two slit experiment, light (or electons, or - more recently - 
neutrons) can show interference effects, a property of waves. The photo-
electric effect shows light to be made up of particles, photons. Both are 
true to an extent.
QED (Quantum Electro-Dynamics), the most successful and accurate theory in 
the history of the human race uses this wave-particle duality. Feynman 
diagrams eg:
\   /
/~~~\
 can be drawn to show an interaction, but the particles can take any route 
and do anything. The particles, however, have a wave-like property called 
phase, which can make two routes cancel (interfere) so that when all 
possible routes are summed, it appears that they move in a straight line.
For more information, I suggest you read a book on Quantum theory or on 
QED.