*Note*  Since trace has several meanings and each can be traced to a slightly different root, I focused on this meaning.  
As a noun meaning a very small amount, trace comes from the Old French noun trace (meaning a footprint or a mark left behind), which is a back-formation from the Old French verb tracier.  

According to the Oxford English Dictionary (OED), the noun trace original meaning emerged as a track, footprint, or mark left by a passing person or animal was first recorded in English around 1380.  The OED defines the figurative sense as "an indication of the presence of a minute amount of some compound; a quantity so minute that it cannot actually be measured first dates to the 1530s.

In storytelling, a "trace" refers to the subtle residues, clues, or echoes left behind by events, characters, or histories. It is the difference between explicitly telling the audience what happened and showing them the evidence, acting as a powerful tool to build depth, immersion, and engagement.  

The Core Roles of Trace:  
Suspense and Foreshadowing: Traces serve as breadcrumbs—such as a torn letter, a lingering glance, or a forgotten item. They hint at a broader narrative or upcoming twists, pulling the audience forward.  
World-Building: Leaving physical or emotional traces (like scars, ruins, or fading traditions) makes a fictional world feel lived-in, real, and anchored by a rich history.  
Active Engagement: Instead of spoon-feeding the plot, a storyteller uses traces to invite the audience to do the detective work. Connecting the dots makes the viewing or reading experience highly interactive and satisfying.  
Thematic Resonance: Traces represent the lingering impacts of themes like trauma, memory, and time, ensuring that past events continue to affect characters in the present.  

By strategically placing traces, storytellers create richer, more resonant, and immersive narratives.