When one searches for evidence to support a theory, such as the theory of evolution, one searches for data that will support the statement of the theory.  Equally important are items that show a divergence, or a separating of two distinct organisms in their development.  Analogous structures are evidence that would fit that definition of divergence.  The source below I listed uses the example of comparing the wing structure of butterflies and bats.  Both these structures are used for the organism to fly, but both are structurally different and therefore, functionally different as well.  Analogous structures may be used to illuminate the chosen path one has determined as the way an organism evolved to its present existence.  It is important to remember when establishing an evolutionary pathway, if you can't proceed in a certain direction because the evidence does not support that pathway, could it be explained in an opposite way?  The use of analogous structures offers that key of explanation, if necessary.

Why are homologous structures evidence of evolutionary relationships?

It helps to know what homologous structures are. They are structures in different organisms that all come from a common primitive ancestor, but they differ in their adaptation and the function thereof.  A good example would be the forearm structure of a bat, a whale, and a human.  They all have the same general structure, all have the same number of bones in the structure, but differ in the function.  The bat uses its wing for flying, while the whale uses its fore flipper for steering its massive body.  The human uses its forearm for grasping and holding objects, as well as providing balance while walking upright.  The anatomical similarities in structure are what provide the evidence of an evolutionary relationship between those three organisms.  They simply had a need to adapt thosae structures for the purpose stated, and the offspring that exhibited those traits were the ones that survived in the game of life referred to as "natural selection".