Is the mysterious footage in the middle of the transition actually the beginning of that same physical clip? (Or maybe the end of the second clip?) This would be the true beginning or end of the video file -- not just beginning or end of the trimmed down event as it appears on the timeline.

If you are at the physical end of a clip (media file), and you b.u.t.t another clip up next to it, then drag and drop a transition, Vegas will automatically create the 1- to 2- second overlap that it needs to create the transition. However, you might not have the extra 1 to 2 seconds head and tail on those clips to create that overlap properly.

By default, when Vegas hits the physical end of a clip, it circles around to the beginning of that clip and continues playing video from there (this is called "looping"). If this is what's happening, you can right-click on the video event on the timeline, click on "Switches", and then disable the "Loop" checkbox. With "Loop" turned off, when Vegas hits the end of that clip it will basically do a freeze-frame of the last frame.

If you untick Looping, you won't see the beginning of the clip during your transition, but you will have a freeze-frame -- which probably won't look right either.

The proper solution is to ovelap the two video events for the duration of the transition and make sure you have enough "head" and "tail" on the two clips to last through the transition.

(Note: I had to spell it "b.u.t.t" above because the forum's ridiculous new "naughty language" scanner thinks b-u-t-t is too vulgar and changed it into "****". I think that's pretty damn stupid.)
(Wow -- even "s.t.u.p.i.d" gets replaced. Note that all replacements show up as 4 "****" -- even if it's a 6-letter vulgarity like s.t.u.p.i.d. But the word right before it was indeed a 4-letter word, beginning with "d'.)