The Giants have been discussing for weeks what to do about Hill, after the news broke that he'd failed another drug test and was facing a third drug suspension in three years. Apparently, they decided another suspension would be too much for them to overlook and they would cut him if/when said suspension was announced. Initially, it appeared Hill was facing a suspension of a year or longer; the fact that it turned out to be only six games seemed to indicate there had been some extenuating circumstances the NFL found plausible. But whatever they might have been, they weren't enough to convince the Giants to keep him around.
It might be tempting to laud the Giants for taking a stand against drugs or bad behavior in general. And if you're a Giants fan and you want to feel that way about your team, go right ahead. But it's worth noting that this was Hill's third drug suspension in three years, so if the Giants really were that hard-line on this issue, it's unlikely he'd have made it this far.
I don't think the Giants were opposed to standing behind Hill and helping him work through whatever issues he might have. What I do think is this has reached a point beyond which the Giants feel they can trust Hill enough to make it worth keeping him on the roster. It's one thing to know a guy has a drug problem and you're working to help him deal with it (which is what the Giants have been doing since signing Hill in 2012). It's another thing, in the cold, business world of the NFL, to keep using a roster spot on a player you can't rely on to be available for a whole season. Whether it's a guy who gets hurt all the time or a guy who gets suspended all the time, it's difficult to invest resources in a player who can't be counted on to play a full season.
Had the Giants decided to keep Hill, who was a good player for them and well liked in the locker room, they'd have done so knowing that future discipline against him was only going to get harsher and more extensive. They already knew he was more likely than not to keep getting in trouble. What happened Monday was they reached the point at which they could no longer tolerate the risk that he might disappear from their plans for a long time. For his first two years with the team, Hill's talent made him worth that risk. As of Monday, in their eyes, it no longer did.