The international order is often called the 'US-led rules-based interntional order'. Russia, China, and some others dislike the first element, of course. The second element refers to the legal, rules-based structure (rather than power-based anarchy, which led to the centuries or millennia of war before the 'order' was created post-WWII). Aggressive international warfare is outlawed, for example; if France and Germany have a dispute, there is no question of violence - they use a legal structure to resolve it, which wasn't always true!
Grey zone activities accomplish illegal things without reprocussions. And therefore they also serve the goal of undermining the international order by demonstrating its powerlessness in these situations. In some ways, it's like trolling.
Russia uses grey zone tactics heavily - for example, they used them to capture Crimea (which was before the clear act of war, their 2022 invasion). They use them to run destabilizing 'grey zone' campaigns throughout the world, including directly interfering in elections. The tactics suit Russia in particular because they cannot compete miltarily with the democratic world.
China uses them too, for example using their 'coast guard' and 'civilian' 'fishing boats' to attack (up to a point) and intimidate ships from other countries in the South China Sea. If China used their navy, it would possibly be acts of war. A Chinese coast guard ship shooting water cannon at a fishing boat, though illegal in international waters, isn't going to start a war. 'Civilian' 'fishing' boats from China blockading access to a reef won't either.
Edit:
Before you look at Russia and China and other Grey Zone actors as miscreants, understand that it's just the normal behavior of 'revisionist' powers - those who want to change the current rules. The current rules serve the interests of the 'status quo' powers, who get all self-righteous about 'illegal' activities.
In a more common situation on HN, think of IP outsiders, who break the 'rules' made by major IP holders, such as DMCA or those extending copyright for decades or restricting access to scientific knowledge - the IP holders want the status quo and call violations 'theft' and the outsiders 'criminals', etc. If the US wasn't a status quo power, they'd be doing grey zone things.
(That doesn't at all justify Russia and China's goals of stealing land, oppressing people's freedoms, and solving problems through violence.)