The Book of Job has inspired much commentary about its presentation of the problem of evil, which is the question of how a God that is allegedly 1) all-powerful, 2) all-good, and 3) all-knowing allow suffering and evil to exist. Job does nothing to deserve divine punishment, yet a series of tragedies befall him as a test.
Ultimately, God replies, but God's answer does not directly reconcile the problem of evil. God speaks of the wonder of creation and God's power, but never as to why job must suffer or why evil is allowed to exist. Essentially, God answers with a non-answer, suggesting it is futile to ask why evil is allowed to exist or why good people are made to suffer and evil people sometimes triumph.
You ask what the Jewish and Christian interpretations of the work's treatment of these themes and ideas, but truth be told, several answers exist among the followers of both faiths. Some of the Jewish faith, like Rabbi Harold Kushner (who has studied and written commentary on the Book of Job), believes God is all-good and all-knowing, but not powerful enough to intervene with humanity's free will.
A common Christian interpretation is that it is unknowable why God allows bad things to happen to good people, but that humans must have faith in God and believe they will be rewarded for their perseverance. As shown in the whirlwind speech, God does not need to justify his ways before mankind, so people should just have faith that goodness will be triumphant in the end.
On the website My Jewish Learning, Robert Seltzer gives another interpretation of the speech, even adding in the idea that reincarnation might solve the problems addressed in the book:
The book reaffirms Job’s trust in God—and God’s trust in Job. In teaching that piety must be unselfish and that the righteous sufferer is assured not of tangible reward but of fellowship with God, biblical thought about justice, retribution, and providence reaches a climax—and a limit.
One alternative that the author of Job did not consider was that the sufferings of the innocent might be compensated in a future life. The problem of theodicy is resolved through just this means in post-biblical Judaism.
In the end, whether or not the book "solves" the problem of theodicy or not is going to depend heavily on one's personal views.