How does increased fluid velocity give rise to low fluid pressure at a point?

Two equations are sufficient to prove this

Firstly equation of continuity A1v1=A2v2 ........A~area v~ velocity

Since you press the pipe its area decreases thereby velocity increases

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Next is bernoullis theorum according to which

h$rho$g +1/2$rho$ $v^2$+ $P1$= constant

Since this sum is constant if one increases other has to decrease. If at a point velocity increases than other point then pressure has to decrease as compared to the other point to ensure the sum remains constant