Why is the polar triangle useful in spherical geometry? We can solve many problems in spherical geometry by using the polar triangle. I am looking for an intuition why (and when) this is easier than working in the original triangle.

Consider the ${fundamental} $ ${equations}$ of the spherical trigonometry, see figure, 
Let $A, B, C$ be three angles of a spherical triangle $ABC$; $a, b, c$, the sides opposite these vertices respectively, then it can be shown that we can obtain three equations describing relationship between them, 
$$\cos a= \cos b \cos c + \sin b \sin c \cos A$$
$$\cos b= \cos c \cos a + \sin c \sin a \cos B$$
$$\cos c= \cos a \cos b + \sin a \sin b \cos C$$
These equations are the fundamental equations since given three quantities, the other three can be computed easily by solving above equations. 
By definition when two triangles are so related that the vertices of the one are the poles of the sides of the other, i.e. the relation of the triangle to another triangle is reciprocal, that triangle is called polar triangle. 
Now for the above figure, let $a',b',c'$ be the sides and $A',B',C'$ be the opposite angles of the polar triangle corresponding to the original triangle $ABC$ such that:
$$a' = 180 - A, A' = 180-a$$ 
$$b' = 180 - B, B' = 180-b$$ 
$$c' = 180 - C, C' = 180-c$$ 
Since the fundamental equations are also satisfied by the polar triangle, using which new relations can be established from the fundamental equations about the original fundamental quantities:
$$\cos A = - \cos B \cos C + \sin B \sin C \cos a$$
$$\cos B = - \cos C \cos A + \sin C \sin A \cos b$$
$$\cos C = - \cos A \cos B + \sin A \sin B \cos c$$
Now to answer your question "why (and when) it is easier working with polar", you can see that in the above derivation, the use of polar triangle is not necessary. They can be directly derived from the fundamental equations by elimination of first $b$ and $c$, then $c$ and $a$, then $a$ and $b$. But the use of the polar triangle gives a more intuitive geometrically assisted, shorter and supposedly more elegant mode of deducing them.  Because of the fact that the sides and the angles of a triangle are respectively supplementary to the angles and sides of the polar triangle, many relations as such can be easily derived using the Polar triangle, hence a triangle and its polar are sometimes called "supplemental triangles."
You can refer to Spherical Trigonometry, Spherical Trigonometry For Colleges for more details.
