Richard Bauckham is the Bishop Wardlaw professor of New Testament Studies at The University of St. Andrews, St. Mary's college. Basically, the thesis of Richard Bauckham's book was first published in the Journal for the Study of the Historical Jesus 5 years ago, and in a nutshell, this is his thesis: The original eyewitnesses to Jesus' life, ministry, death, and resurrection were the ultimate origins of many of the traditions found in the Gospel narratives and played a vital role in their transmission up until the time of the composition of the Gospels.

These are the 18 chapters in Bauckham's Study
1: From the Historical Jesus to the Jesus of Testimony
2: Papias on the Eyewitnesses
3: Names in the Gospel traditions
4: Palestinian Jewish names
5: The Twelve
6: Eyewitnesses "from the beginning"
7: The Petrine Perspective in the Gospel of Mark
8: Anonymous persons in Mark's passion narrative
9: Papias on Mark and Matthew
10: Models of Oral tradition
11: Transmitting the Jesus traditions
12: Anonymous tradition or Eyewitness testimony?
13: Eyewitness memory
14: The Gospel of John as Eyewitness testimony
15: The witness of the Beloved disciple
16: Papias on John
17: Polycrates and Iranaeus on John
18: The Jesus of Testimony

Bauckham in these 18 chapters presents cogent, powerful, and above all useful arguments for the reliability of the Gospel portaryal of the Jew from galilee. The first argument he presents is that Papias' prologue, preserved in Eusebius, contains information that he assembled his 5 books called The Exposition of the Logia of the Lord, and that he personally had access to traditions rooted in eyewitness testimony. His phrase "a living and surviving voice" is very similar to the opening statement in Luke's Gospel, and conforms to other Ancient historiographic preference for living oral sources rather than written sources. This similarity shows that Luke had access to traditions extensively rooted in eyewitness testimony, and hence the rest of the Gospels likewise had similar testimony to back up their veracity.
ADDENDUM: After contemplating and re-reading Bauckham's 2nd chapter on Papias and the Eyewitnesses, I think a second very cogent argument this gives for the Chrisitans is that Papias' claim to have access to traditions passed down by "the Elders" like John the Elder or Aristion of Philipp the Evangelist shows that the witnesses were by-and-large passing down their stories. Even when Papias wrote (90s AD), long-lived disciples and witnesses still told their tales. This is strong evidence that the Gospels surely would have been rooted in eyewitness testimony.
The second argument comes from his second chapter. In this chapter, Bauckham discusses a pattern of name-dropping we observe in the Synoptic Gospels. In Mark, we see some named characters change to anonymous characters in Matthew and Luke, but we never once observe an anonymous character in Mark gain a name in Matthew or Luke. Couple this with the fact that there are many named characters where the names are simply unecessary (Such as Simon of Cyrene and his sons), or anonymous characters where a name may be necessary (Such as Cleopas on the road to Emmaus), seem to give a nice cumulative case for this phenomena being best explained by the named figures being the very persons who passed down the traditions about these events in the life of Jesus
The third argument which was marginally helpful was his discussion on Palestinian Jewish names. Richard Bauckham makes use of Til Ilan's Lexicon of Jewish names and shows how the personal names in the Gospels conform well to known Palestinian and Galilean naming practices. I dont remember if Bauckham pointed this out in his book, but this would seem to indicate an Aramaic substratum to the Gospels. If the Gospel tradition has an aramaic substratum, this would attest to its reliability and connection to much older traditions, strengthening the case for eyewitnesses involved in the origination and transmission of the Jesus traditions
The fourth argument I found to be very helpful was Bauckham's case that the careful preservation of the list of the twelve apostles indicates that the Twelve disciples, with the exception of Judas Iscariot, were an official body of witnesses who passed on the Jesus traditions. This, again, further reinforces Bauckham's case for the involvement of eyewitnesses in the origination and transmission of the Jesus tradition
The fifth argument I shall repeat is Bauckham's discussion of the literary device of inclusio and its use in the Gospels of Mark, Luke, and John. Basically, Bauckham shows how a good portion of Mark's Gospel sort of revolves around Peter, Luke's gospel revolving around the women disciples, and John revolving around Peter, and around that inclusio (I am using "inclusio" and "revolution" interchangeably as I think it would be easier for my audience to visualize this device) there is an inclusio around The Beloved Disciple. Bauckham's discussion on Porphyry and Lucian's use of a similar device in their works to show their own primary witness lends strong evidential support to Mark, Luke, and John revealing eyewitnesses as their main sources.
The sixth argument I will provide is expressed in 2 chapters of Bauckham's work, which is an argument for the traditional authorship of Mark's Gospel. First, Bauckham discusses how the Gospel of Mark is essentially a Petrine Gospel, in that it is more centered around Peter than any of the other Gospels, and this provides strong internal evidence of Peter being the primary witness behind Mark's Gospel. Bauckham then discusses the external evidence from Papias of Hierapolis and its attestation of Markan authorship

I will stop for now and simply recap. In this book, Bauckham provides an incredibly scholarly and compelling case for the reliability of the Gospel tradition by showing its essential origins on eyewitness testimony.