Why do rivers near active faults erode faster than rivers farther away?

Answer: The fault breaks up rock across a broad area, leaving weaker rock that rivers erode faster.

The fault breaks up rock across a broad area, leaving weaker rock that rivers erode faster. ✓ — Correct. An active fault does more than make one clean crack. It breaks the rock across a broad area, and that broken rock is easier for rivers to wear down. So channels near the fault can cut down faster than channels farther away.

The fault heats the river water, and warmer water cuts into the riverbed faster. — Wrong. This makes water temperature the key, but a fault does not add enough heat to river water to change erosion here. The important change is in the rock: fault movement breaks it and makes it easier to remove. That mechanical weakening is what lets nearby rivers cut down faster.

The fault is only a thin crack, so it changes just the rock right at the crack. — Wrong. It is true that a fault can look like a single narrow line, especially on a map. But the rock around that line is also cracked and crushed over a broad area. A thin crack by itself would not make rivers across a whole region erode faster.