And the flood was forty days upon the earth; and the waters increased, and bare up the ark, and it was lift up above the earth. (Genesis, Chapter 7: 17)
What does "bare up" mean in here? And why?
This is an archaic construction; here, bare is used as the past tense of bear, to carry (or, in this case, to lift).
This is from the King James version of the Bible, known for many archaic constructions and translations. Another version says:
For forty days the flood kept coming on the earth, and as the waters increased they lifted the ark high above the earth.
And the waters became powerful, and they increased very much upon the earth, and the ark moved upon the waters.
In your version "bare" is an old and unused spelling of "bore", the past tense of the transitive verb, "to bear", meaning "to carry". Example:
Although she was elderly and the child was no longer small, she bore his weight easily as she rocked him to sleep.
According to Webster's Ninth Collegiate Dictionary, "bare" is the archaic past tense of "bear". In turn "to bear" means to carry. So, the expression "bare up the ark" means that the water supported the weight of the ark.
Note that in the expression "bare up" the word "up" does not mean "upward". It simply means that the waters did not let the ark sink. Compare: "He is falling, hold him up." There are two verbs in this sentence. The first (bare up) means that the waters bore the weight of the ark, the second (lift up) means that the ark rose as the waters which were holding it up rose. This corresponds with the Hebrew original as explained by user @LukeSawczak.
Today we do not use "bear up" in the literal sense of supporting weight. But, we do use it in the figurative sense of enduring difficult situations without giving up. Such uses are intransitive (do not name a direct object). For example: "He bears up well under stress."