Short answer:
In the context of the question, I believe the answer was just saying that running is far more exhaustive than walking. Thus walking will leave you less drained than running will.
Long answer:
There are two aspects to look at, the effects of running on your energy levels and the effects of running on your body.
Regarding energy levels, the human body has various energy stores that are used to produce the exertion needed for moving. It takes more energy to move a single object over a set distance at a fast rate then it would to move the same object over the same distance at a slow rate.
Then there's how running effects the rest of the body. Mainly the muscles and bones. Depending on how fast you run, the muscles are going to break down and damage a little. Then they heal and become stronger next time. This is a necessary cycle for adaption, but there is this recovery time immediately after a run where your slightly weaker than when you started. The bones during the run also act like shock absorbers taking in the impact of each stride. Likewise, bones become hardened over time, but there is still a recovery period. Walking, in contrast, isn't as intense and does not have such a recover period.
All of this is relative to the fitness level of the individual though. Most people can probably walk for ten miles given the right motivation. Those that don't walk often will feel exhausted because they have not adapted to the work. Those that walk often will feel great. Likewise, an ultra-marathon runner could run ten miles and feel great afterwards.