The sense of this formula is clear. It means be patient with me, be tolerant/lenient. Don't be too harsh on me.
But how can a verb as "to bear" develop the meaning of to be tolerant?
"To bear" is an old verb with connection to Latin ferre and Greek pherein.
"To bear" has split up into several meanings such as "to carry", "to give birth to a child", "to endure", "to be directed towards" and similar things which can semantically be understood from the basic meaning to carry or from Latin ferre and Greek pherein, but no meaning shows a semantic path to "Bear with me".
Best Answer
From the Phrase Finder:
The expression is quite old and can be found in Shakespeare's works: