Why does the pressure of fluid (incompressible) decreases with increase in its velocity? I don't want an explanation with the help of Bernoulli's...

Bernoulli's equation is the simplest way to understand this. This equation is a result of conservation of energy in a flow, and will simply mean that (in absence of a change in datum level) potential energy will be converted to kinetic energy of fluid. However, if you do not want to use an explanation based on this equation, try to think about the situation this way. When the area of flow changes (say decreases), there is less area and according to continuity equation (A1V1= A2V2), flow velocity will increase (to accommodate the same flow through a smaller cross-section). However, if we discount all other forces, how can this be? The flow is caused by a gradient, in this case, pressure gradient causes the flow. Thus, the pressure must decrease across the cross-section, causing an increase in the velocity of fluid flow.  

Hope this helps.