Compare the two reactions:
A
P
A + A
P
In the second order reaction, a collision is required between two A molecules in order for one product conversion to take place. As A is converted into P, there is less A in the reaction vessel, so the probability of a collision becomes lower and lower. The reaction rate is very slow as the concentration of A becomes vanishingly small.
In the first order reaction, no collision is required. The probability per unit time of a successful conversion event still decreases as time elapses, because there is still less A in the reaction vessel at later times, but this drop in the reaction probability is not as severe as it is where a collision is required.
The amount of A left in the reaction vessel per unit time is inversely dependent on the reaction probability per unit time. This is why the concentration of A drops faster in the first order case than it does in the second order case.