The proverb refers to water ways that may be man-made or natural bridges. As such a stream or a brook may look like a small river that can be crossed easily. However if one is crossing such a stream there is still need to tuck up their clothes so that they do not get wet or hinder their ability to cross the stream. Hence one must not misjudge the challenge ahead of them but must always maintain a humble disposition in order to meet the challenge effectively. Our elders were teaching us the lesson that we should not underestimate a challenge because it does not appear as severe, but we must accept that every difficult situation requires humility to assess and address the situation adequately. An arrogant disposition usually underestimates a challenge.
The proverb is used to warn people that in approaching difficult situations, regardless of how small they may appear, one must humble themselves and respect the challenge. It is particularly useful to those who are normally of a pompous or egoistic disposition as a admonishment that their attitude will not prevent then from effectively assessing and addressing the challenge as they are bound to underestimate the challenge.