Intro

Recall that BFS finds the shortest path by expanding in layers. Layer $1$ ($1$ step away), then Layer $2$ ($2$ steps away), etc. This assumes that "$1$ step" always equals "distance $1$".

But look at this graph: $A→B$ (Distance $10$) $A→C$ (Distance $1$) $C→B$ (Distance $1$) BFS will see $A→B$ immediately ($1$ step) and declare the distance is $10$.

But the path $A→C→B$ is distance $1+1=2$. BFS fails here. You need something smarter.