How does Grand Battleplan save on manpower?
Much like superior firepower, indirectly.
Superior firepower doesn't "save on manpower". What it does is increase soft attack and give a flat buff to defence. Soft attack means the enemy will generally take more casualties than you, and defence means that you take fewer casualties when defending.
It works well with the classic anti-AI strategy of sitting back on a big defensive line and letting the enemy attack you until they run out of equipment and manpower, and while you can still do that it's not as reliable as it used to be.
GBP, instead of giving flat buffs to soft attack and defence, increases max entrenchment, planning bonus and breakthrough. The first two translate into situational boosts to attack and defence which are generally actually higher than those provided by superior firepower, but which require certain conditions to be met. A fully entrenched division with grand battleplan will usually have higher defence than a superior firepower division, but a division which has just moved will be more vulnerable. Similarly, GBP divisions can have very high attack and take fewer casualties, but only when carrying out a fully prepared plan.
Then there's breakthrough, which works like defence but when attacking. It means your divisions will generally take fewer casualties when attacking. Even basic infantry divisions can have respectable breakthrough with GBP, and since attacking tends to produce more casualties this can result in fewer casualties, especially when combined with the planning bonus.
In short, it's a really strong and very underrated doctrine, probably because a lot of players see it as playing like an AI (which is not true, you can and probably should still micro with GBP).