The term digging is in reference to the amplification of a trough. When a trough becomes more amplified (more meridional and larger) it is said to be "digging". A trough will become more amplified if a strong jet streak is in the entrance region (to left of trough axis) of the trough. When a trough amplifies, the upper level flow associated with it becomes more meridional and the influence of the trough moves into lower latitudes.
The links below show the relationship between jet streak position and how trough changes over time
The term deepening implies:
(1) surface pressure of the low is decreasing over time within the center of the low
(2) the surface low has more closed contours of isobars around it from the last analysis / forecast prog
(3) heights are lowering aloft above the low (height falls)
(4) the low is becoming closed or has more height contours encircling it on the upper level charts.
The opposite of digging is lifting and the opposite of deepening is filling.