Q:

Everyone knows that there is only one nonsingular conic tangent to five general lines in $\mathbb{P}^2$

This is a statement in Fulton's Introduction to intersection theory:
''there is only one nonsingular conic tangent to five general lines in $\mathbb{P}^2$ ''
He implied that this is clear by duality, for example. 
Anyone could Please give me a more lucid explanation? 

A:

As a complement let me emphasize that a more precise and more  elementary statement, avoiding the  complication of the term "general line", is:   
Given five lines in $\mathbb P^2$ , there exists a unique smooth conic tangent to these five lines if and only if no three of these lines have a common point. 

A:

The term duality here refers to the dual projective plane $(\Bbb{P}^2)^\ast$, definined as the set of lines of $\Bbb{P}^2$.
Explicitly, a line $ax+by+cz=0$ in $\Bbb{P}^2$ becomes the point $(a:b:c)$ in $(\Bbb{P}^2)^\ast$.
How do you dualize a smooth curve $C$ in $\Bbb{P}^2$ then?  You just define the dual curve $C^\ast$ to be the set of points of $(\Bbb{P}^2)^\ast$ corresponding the tangent lines to $C$. 
Notice that $C^\ast$ may be singular: for example if $C$ admits a bitangent line.  Also, the degree will not be preserved in general. 
As Ted Shriffin suggests in the comments, however, the dual of a conic will be a conic.  Moreover, it will be smooth ! ( try to see why). 
Finally, the statement
there is only one nonsingular conic tangent to five general lines in $\Bbb{P}^2$
is equivalent to its dual statement:
there is only one nonsingular conic passing through five general points in $(\Bbb{P}^2)^\ast$
If you identify $(\Bbb{P}^2)^\ast$ with a standard projective plane, the dual statement is just the classical assertion that there exists a single smooth conic passing through 5 general points in projective plane.  Are you fine with this assertion?  Notice that an idea for proving it is contained in KReiser's comment above.