True-crime-bragging is a new coinage, a one-off phrase. It's meant to evoke the recently popularized humblebrag
: to make a seemingly modest, self-critical, or casual statement or reference that is meant to draw attention to one's admirable or impressive qualities or achievements
This definition is not especially clear, and people use the phrase pretty inconsistently.
To my mind, a humblebrag is a statement sounds like it's meant to evoke sympathy, but which is actually meant to provoke jealously, or to show off.
Here's a little illustration: imagine a person has just paid someone to build them a large and unusual house.
If the person wanted to brag, they would say: "This house was so large and unusual that only someone as rich as me could afford to pay for it."
To humblebrag, the person would imply the same thing (only I'm rich enough), but would phrase the statement differently, so that the listener might think we're supposed to feel bad for them: "It's so difficult to build a house like this. Getting all the permits was a nightmare. I wish I had built something smaller and simpler."
This one-off phrase should be understood as a specific form of humblebragging - one that relates to the "true crime" genre.
In the example text, Beth makes a statement which seems like it's meant to evoke sympathy: "I'm sad/scared/upset because the person he killed was my best friend."
But the statement also, or actually, invites envy: "We're all interested in this exciting murder case - I deserve special attention because I knew the person who was killed."