A veto is usually an all-or-nothing instrument. In most states the governor holds a narrower and more powerful version that applies to parts of a bill.
The mechanism is partial rejection
Presented with an appropriations bill, a governor with this authority may sign it while striking specified items, which then do not become law while the remainder does.
The legislature can usually restore a struck item, but only by assembling the supermajority required to override, which is far harder than passing the item originally was.
The practical effect is that the governor becomes the final editor of the budget rather than a party who must accept or reject the legislature's product entire.
It disarms the standard legislative tactic
Legislatures attach unpopular provisions to essential bills precisely because a governor will not shut down government to reject one item.
Where the line-item veto exists, that tactic fails, because the governor can remove the addition and enact the rest. This shifts leverage decisively toward the executive.
Legislators respond by writing conditions into general statutory language rather than into appropriation lines, since the authority usually reaches spending items and not substantive law.
The scope varies considerably between states
Some constitutions limit the authority to striking whole appropriation items, while others permit reducing an amount without eliminating it, and a few have permitted striking individual words.
Broader versions have allowed governors to change the meaning of a sentence by deletion, producing text the legislature never voted on and prompting amendments to narrow the power.
Courts in several states have been asked to draw the boundary, generally holding that the authority is negative — a governor may subtract but may not create new provisions.
Anticipation matters more than use
Counting vetoes understates the effect, because legislative drafters write with the veto in mind and often omit provisions they expect to be struck.
Negotiations over budget language therefore happen before passage, with the governor's staff involved, which is why the enacted bill frequently requires few actual strikes.
A governor who uses the authority heavily is usually one whose legislature is controlled by the other party, since a friendly legislature has already accommodated the objections in advance.
The national comparison is instructive
The federal executive does not hold an equivalent authority over ordinary appropriations, and attempts to create one by statute have run into constitutional objections about how a bill becomes law.
State constitutions were largely written or amended later and incorporated the device deliberately, which is why the same office has different powers at different levels of government.
Anyone reasoning from national practice to state practice will therefore misjudge how budget disputes resolve, because the executive's position in the bargain is fundamentally stronger in most states.