Because it allows the President to pass a bill that is fundamentally different from the one Congress sent him.
To go back to civics class 101, the House or Senate proposes a bill that, upon passage in that house, gets sent to the other. The other then makes amendments or writes is own version and passes it back. The bill goes back and forth until both houses agree and only then does it get sent to the President. At that point a clock starts ticking. The President only has 10 days (Sundays excluded) to either pass or veto the bill (with some additional caveats that are irrelevant at the moment).
The Line-Item Veto act of 1996 allowed the President to pass the parts he likes, and pass just his objections back to Congress. This seems like a very small change, but in actuality it completely changes the power dynamic between the President and Congress.
Under the new law, when the President used the veto, another clock would start for Congress to either override the veto, or pass changes that would again get sent to the President for approval. In either case, however, Congress would only be able to discuss the objections the President sent back to them. They would not be able to do anything about the parts he passed, as those would now be law. That's a bad thing... Like, a really bad thing.
Let's use an example here. Let's say that in a rare case of bipartisanship, Congress passes a bill offering free college tuition to all Americans at any public university of their choice. To pay for that grant, Congress also includes a major tax hike in this very expensive bill. Of course, free college tuition is politically popular, but a tax hike is not. So the President passes the free tuition part and vetoes the tax part.
Now we have a problem. The President just vetoed the only funding source that Congress could agree on. Now they have to go back to the drawing board and find another way to pay for it, because free tuition is now the law and the only way around that is to pass a repeal bill (which the President would have no incentive to sign). And to top it off, they only have a limited window of time to do it.
In effect, the 1996 act made the President a de facto 3rd house of the Legislature (which is not the President's job), with the added power to "lock out" portions of the bill from being discussed further (something neither house of Congress has).
So I hope that everyone who reads this knows now how colossally bad an idea the line-item veto is in the American system. The Supreme Court made the right call to overturn it.
Also let's not forget that such a law represents a fundamental change to Article I of the Constitution, which requires a constitutional amendment; that can't be done through the ordinary lawmaking process.