Biblical Criticism, in particular higher criticism, deals with why and how the books of the Bible were written; lower criticism deals with the actual teachings of its authors. The word "criticism" must be one of the all-time least appropriate religious terms. Theologians do not engage in actual criticism - at least as the word is commonly understood. They analyze the Bible in order to understand it better.
Biblical criticism originated with anti-Christian writers who valued reason and logic over faith and revelation. Their goal was to discredit and ridicule the Bible and Christianity. Their analysis techniques were picked up by some liberal theologians and initially used to explain away and discount Biblical accounts of prophecy, miracles, personal demon infestation, etc. Finally, even mainstream theologians began to use biblical criticism to determine:
"which are the most reliable and trustworthy texts" of the Bible.
how are various of its books related to each other
who were its authors
when were they written
which passages are of real events; which are myth, legends, folklore, etc. Which are religious propaganda, etc.
"...what is the relationship of these sources to other oral and written material of the time?"
Theologians use biblical criticism to date when various books in the Bible were written. This helps them:
detect when, in the early Christian movement, various beliefs (like the virgin birth) first arose.
detect when animosity against the Jews developed in Pauline Christianity
determine whether 1 Timothy, 2 timothy, and Titus were written by Paul or by unknown persons in the 2nd century CE, 35 to 85 years after Paul's death.