Firstly, safety.
Stepping or leaning on a rotten log is not fun if you end up going through it ( Dynadin did it when Geocaching and ended up with a large number of scratches down one arm, and it could have been worse!)
- there could be a big hole underneath
you can't see.
Also logs which are wet or covered in moss or mildew can be slippery, which may cause you to fall.
This also prevents damage to the ecosystem (lots of animals make homes in rotten logs).
Secondly, noise.
Breaking through bracken or sticks etc when you can step over them could cause your prey when hunting to flee, if you can prevent this by simply stepping over said stick, log, etc, then you may as well do so and save yourself some aggravation.
outdoors.stackexchange.com/users/3313/Aravona
It means to be mindful of what you do, how and where you move, what you do to your environment and so on.
Aside the topics mentioned by @Aravona (security, noise), it may mean to avoid unintended killing of small animals or plants, or to generally avoid leaving any marks in nature, or actually wherever you move.
Sound advice not only for hunting, I'd suggest.
outdoors.stackexchange.com/users/9349/AnoE
This statement is one of those pieces of wisdom that sounds profound, but isn't.
Following this advice can lead you into trouble as easily as it can help you avoid trouble.
The other answers have illustrated the latter.
However, before you step over something, you better know what is on the other side.
Stepping on a flattish rock is better than stepping over it into a mud puddle or onto a clump of flowers or that snake.
outdoors.stackexchange.com/users/6411/ab2
I have always heard this in a hiking context, where it is fatigue and safety related.
There is also a 'second verse' in the version I was taught: dont step on something you can step around.
It is basic common sense, but combined into a mantra since after a day (or multiple) on the trail with a pack, fatigue does set in.
Its easy to justify in your mind the short cut to step on the rock instead of stepping over because its a small rock, or over instead of around the fallen tree to save 4-5 steps.
Too often these sorts of short cuts lead to injury or worse, and they usually occur when tired, or in a rush.
outdoors.stackexchange.com/users/11045/irrogical
The canonical reason for this (in a hiking context, where noise is not an issue but fatigue can be) is that it saves effort.
Think about the physics -- when you step up on to a log, you are doing some work to lift your body and gear up there.
Unless you then jump off the log (which could be dangerous), you are also doing some work to slowly lower yourself to the ground level.
Alternatively, when you step over the obstacle and put your foot back down on the other side at the same level, you aren't really expending any more effort than normal walking.
(setting aside that you may have to lift your foot a little higher so you don't trip!)
I have heard this explaination multiple times from various weathered hikers, but the first was probably something to do with Cub Scouts
-- I was able to locate a reference in the Scout handbook, and find it highly likely that this is the reasoning behind Mr. Bear's (good) advice; possibly also the source.
Boy Scouts Handbook by Boy Scouts of America, 1911 edition, pg.
145, http://www.gutenberg.org/cache/epub/29558/pg29558.txt : "Third, observe these two rules given by an old woodsman: (1) Never walk over anything you can walk around; (2) never step on anything that you can step over.
Every time you step on anything you lift the weight of your body.
Why lift extra weight when tramping?"
outdoors.stackexchange.com/users/11044/jkf
Equal expressions would be "Leave no trace."
and the sentiment of "pack it in, pack it out."
Do not be needlessly invasive on the environment.
If the place is pristine when you get there, it should be when you leave.
outdoors.stackexchange.com/users/11042/dlb
Fred Bear’s 10 Commandments of hunting state as the first principle:     Don't step on anything you can step over    What does he mean with this statement? I don't really get what's the problem with "stepping on something".
"Don’t step on anything you can step over." - What does this mean?
outdoors.stackexchange.com
<hunting><walking>
outdoors.stackexchange.com/questions/13495
outdoors.stackexchange.com/users/8060/OddDeer
Firstly, safety.  For example, logs which are wet or covered in moss or mildew can be slippery, which may cause you to fall. It also means to be mindful of what you do, how and where you move, and what you do to your environment, to avoid unintended killing of small animals or plants, or to generally avoid leaving any marks in nature, or actually wherever you move.  This is about avoiding injury, saving energy and respecting your environment.
This is in order to avoid injury and save energy. This is about respecting your environment.