It's basically a matter of definition.
I think you can look at it this way: The field between a positive plate and a negative plate is said to be directed towards the negative plate (the direction in which a positive charge would move) but the positive plate has a positive potential with respect to the negative plate i.e. voltage increases in the opposite direction to the direction of the force on a positive charge.
you see I have this question where youre give an equation for potential deference of 10-5z^2, so E= -10z right?, anyway there is a field on one side of the slab (slab is the material with a charge) and it asks about EF on both sides, ie what is EF on right hand side and left hand side. As a hint there is "watch the sign" statement. So I am really confused
If you want a simple explanation for the sign, the field points towards the negative side and away from the positive. i.e. a negative slope as distance towards the positive increases - hence the negative sign.
Note. A field isn't "emitted".