Regarding the second half of the book: The 1st person sections (ch.7-12) internally claim to be written by Daniel in fully 18 separate verses. We believe that the author is being honest about his identity.
Ancient Christians and Jews were certainly not unaware of the existence of forgeries and people writing under a false name, but they decided to include the book of Daniel in the canon while rejecting many other books which claimed authorship by prophets and patriarchs. Therefore, they must have had some reason to believe that the book of Daniel is genuine; it's quite possible that the decision to canonize the book was made while Daniel was still around, in which case verifying authorship would have been easy. In any case, there is a very old tradition that the book is not a forgery.
The reason why Christians continue to identify the book's author with Daniel (at least for the last six chapters) is that conservative Christians tend to value tradition over modern scholarship, especially when the modern scholarship is working on scanty evidence. The ones who decided to include the book in the Bible must have had some evidence for doing so, even if we don't know what that evidence was.
The evidence for the late composition date of the second half of Daniel comes largely from the fact that it appears to contain accurate prophecies of historical events from Alexander the Great until Antiochus IV Epiphanes's desecration of the temple in 167 BC, but not about Epiphanes's death; therefore scholars conclude that these passages were written during Antiochus Epiphanes's lifetime.
Note that this logic assumes that predictive prophecy is not possible, an assumption which Christians reject. Without making that assumption, the evidence for a late composition date becomes very paltry. It is not reduced to nothing but it becomes small enough that the evidence of a very old tradition is sufficient for most Christians to follow the traditional view.
Regarding the first half of the book: The idea that the book as a whole was composed by Daniel comes from the assumption that it is written by a single author. There isn't really a reason a priori to suspect that ch.1-6 are written by the same author as ch.7-12, but the ancient readers generally made the assumption of a single author for any book, unless there was a very compelling reason to suppose otherwise. For example, the anonymous Psalms would often be attributed to David (who authored almost all the Psalms that have an attribution); a tradition which the Bible itself follows in Acts 4:25.
Of course, it would not threaten the integrity of the book to suppose that ch.1-6 were written by somebody else. But there is no great reason for doing so, and it appears like ch.7-12 were added later, which implies that at the very least Daniel read and approved the first half (assuming that the later chapters were, in fact, written by Daniel).
The testimony of Jesus: For Christians, we also have the evidence that Jesus believed the book to be genuinely prophetic (he cites it Matt. 24:15 and Matt.26:64), and we believe that Jesus is omniscient.
