Real-world applications:
Snow plowing / Street sweeping: Visit every street at least once.
DNA sequencing: Reconstruct a DNA sequence from short fragments (k-mers). Each fragment is an edge in a De Bruijn graph.
Network routing: Visit every connection in a network.
Circuit design: Draw a circuit without lifting the pen (visiting every wire once).
Puzzle solving: Certain puzzles (like drawing a figure without lifting the pen) are Euler path problems.
When you see "visit every edge exactly once," think Euler paths.