The point of the sparring program is to help Neo "free his mind".
Breathing is an involuntary function of our bodies that helps us keep alive.
The human mind associates "breathing air" directly to "being alive".
We know if anything stops our breathing of air for even a few minutes, we will die.
(example struggling to surface for air in a swimming pool, this is scary for everyone)
In the simulation program, or the matrix, the human mind continues with to perform involuntary actions like breathing, but this is in the mind. For example, in our dreams we continue to breathe, it is but natural and the only thing we are used to doing. But is it required to stay alive in the dream? No.
Morpheus particularly asks this question to Neo to shake the very foundation of this existence, to get him "to believe" that he is in a simulation. Neo technically doesn't need to breath in a simulation. That need is only in his mind. Morpheus is getting Neo to let go of the real world "rules" to help him realize that nothing from the real world holds good in the simulation.
Once that happens, Neo is able to move faster and defeat Morpheus.
But that is just one small step, he still fails the jump test right after :)