I believe that the example http://getbootstrap.com/css/#grid-responsive-resets is incorrect and does not illustrate the problem.
The example http://getbootstrap.com/css/#grid-responsive-resets does not visualize the problem.
your columns are not entirely clear, since one is higher than the other
example without clearfix:
<div class="row"> <div class="col-xs-6 col-sm-3" style="background-color:red;height:40px;">.col-xs-6 .col-sm-3</div> <div class="col-xs-6 col-sm-3" style="background-color:blue;">.col-xs-6 .col-sm-3</div> <div class="col-xs-6 col-sm-3" style="background-color:green;">.col-xs-6 .col-sm-3 (left)</div> <div class="col-xs-6 col-sm-3" style="background-color:yellow;">.col-xs-6 .col-sm-3 (right)</div> </div>
On an extra small (xs) with the first column (red) above, and the second (blue) will call the third (green) column with the right side of the first.
without clearfix

with clearfix
<div class="row"> <div class="col-xs-6 col-sm-3" style="background-color:red;height:40px;">.col-xs-6 .col-sm-3</div> <div class="col-xs-6 col-sm-3" style="background-color:blue;">.col-xs-6 .col-sm-3</div> <div class="clearfix visible-xs"></div> <div class="col-xs-6 col-sm-3" style="background-color:green;">.col-xs-6 .col-sm-3 (left)</div> <div class="col-xs-6 col-sm-3" style="background-color:yellow;">.col-xs-6 .col-sm-3 (right)</div> </div>

Col - * - 12
The problem arises when adding 12 columns in a row, and one of these rows should be 100% (col - * - 12).
Consider this situation:
On the large grids you want:
1 | 2 | 3
On the xs grid you want:
1 | 2
3
You can do the above:
Without clearfix: <div class="container"> <div class="row"> <div class="col-xs-6 col-sm-4">1</div> <div class="col-xs-6 col-sm-4">2</div> <div class="col-xs-12 col-sm-4" style="background-color:red;">3</div> </div> </div>
A red background will indicate that the last column will overlap the first. Adding clearfix will remove this background:
With clearfix: <div class="container"> <div class="row"> <div class="col-xs-6 col-sm-4">1</div> <div class="col-xs-6 col-sm-4">2</div> <div class="clearfix visible-xs"></div> <div class="col-xs-12 col-sm-4" style="background-color:red;">3</div> </div> </div>
Results:

The specified overlap will be caused by the fact that the col-12-* classes do not have a float to the left, see also: https://github.com/twbs/bootstrap/issues/10152 p>
At https://github.com/twbs/bootstrap/issues/10535 you will find another illustration. This script shows how clearfix solves the problem. Note <div class="col-sm-3"> there is no col-12-* . By default, the xs grid columns are 100% and do not have a float, so col-xs-12-* will act just like a non-class in the xs grid.