More than a running total
It is easy to treat the count as a simple tally that ends an at-bat once it reaches four balls or three strikes. I think that undersells it badly. At any given moment the count describes exactly how much risk each side can afford to take, and both players are constantly recalculating around that fact.
A pitcher ahead in the count can afford to throw a pitch outside the strike zone, tempting a hitter who now has to guard against being punished for taking a good pitch. A hitter ahead in the count can afford to be selective, forcing a pitcher who cannot afford to fall further behind into throwing something more hittable.
Leverage, not just arithmetic
Once you watch the count as leverage rather than arithmetic, individual pitches start to make more sense. A borderline pitch taken for a strike in one count is a minor event; the same pitch in a different count can flip the entire tactical balance of the at-bat.
This is, I think, the single most useful habit a newer viewer can build. It costs nothing and it is available on every single pitch of every single game, which is more than can be said for most of the sport's finer details.
A habit worth building
None of this requires new information. The count is displayed on every broadcast and every scoreboard; the only thing missing is the habit of asking, on each pitch, what each side can and cannot afford to do given the number showing. That question turns a string of pitches into a negotiation with a clear internal logic.
It will not make a slow game move faster. But it makes the stillness worth paying attention to, which was always the actual goal of watching more closely in the first place.