How is this quote from Hamlet related to political unrest in Shakespeare's time, and,if so,how has he used this quote to target his audience? I...

In the above quote from Shakespear's
Hamlet,Hamlet reflects upon the character of Fortinbras, who is
Norwegian.  He is the son of Fortinbras, whom Hamlet's father, King Hamlet, defeated and
killed for "a little patch of ground."

Fortinbras serves as
the most important foil to the dilatory Hamlet.  For, as Hamlet notes, he is willing to
"expose what is mortal and unsure"--his life-- for "an eggshell"--the battle which he
may easily lose.  For, while their situations are similar in that their fathers were
killed, unlike Fortinbras who acts upon filial duty, Hamlet is given to excessive
self-debate and procrastination. Described by Hamlet as "a delicate and tender prince"
(IV,iv,48), Fortinbras is easily incited to fight in the cause of national pride or
family duty.

With Fortinbras as a character of national
pride, Elizabethan audiences may well have perceived some parallels between the
Danish and the English court.  While the Spanish Armada had been defeated in 1588, there
yet existed the potential of a renewed invasion attempt, just as Denmark fears an
invasion attempt.  In addition, in England, as in Hamlet, anxieties
regarding royal succession also exist.  One critic named Kurland in "William
Shakespeare's Tragedy as a Political Tragedy Rather than a Political Tragedy," contends
that there are echoes of Elizabeth anxiety over succession, accompanied by fear of
intervention just as occurs in
Hamlet:

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...Shakespeare's audience would have been
unlikely to have seen in Hamlet's story merely a private tragedy, or in Fortinbras's
succession to the Danish throne an unproblematic restoration of
order.

Thus,
it does seem that Shakespeare's possibly play alerts his audience to contemporary
issues. Again, the old question of "Does art imitate life, or life imitate art?"
arises.