In a steady state, amount of charge moving through a conductor per time (i.e., current) must be constant in a steady state because otherwise it would be building up or disappearing; charge doesn't appear out of nowhere or vanish (it's conserved).
You can consider flowing charges to be like continuously flowing water: as water flows through a channel, the mass per time crossing some point must be constant, or else the water would be pooling or leaking.
Note that speed of individual charge flow is not the same as
current. Current, 
, is amount of charge flowing past some
point per time. You could have lots of charge flowing slowly or less
charge flowing quickly to get the same current (charge per time).
In the flowing-water analogy, current is analogous to 
, mass
per time flowing past a point. You could have less mass flowing
faster (just as water flows fast through a small opening) or more mass
flowing faster (just as water slows down going from small to large
cross-sectional area channel). You may remember the ``equation of continuity'',
from your mechanics course.
We'll talk more about this next class when we discuss Kirchoff's laws.