The cessation of nuclear testing by the United States and Russia was meant as a measure to indicate a deliberate cessation of the nuclear arms race in the post-Cold War period. Halting nuclear testing essentially halted warhead development, and so it "froze" warheads in a particular state that both countries could live with.
After this, there were pushes to halt all nuclear testing, and even the creation of a Comprehensive Test Ban Treaty. However the CTBT never went into force.
The other nuclear nations that were testing by the 1990s included the UK, whose nuclear connections with the USA were deep enough that their programs were essentially tied together, as well as France and China. Both France and China continued testing into the mid-1990s, before voluntarily stopping. I don't know the interior thinking of their governments, but generally speaking this was seen as a statement of their sense that their warheads were of adequate quality to guarantee their security.
India and Pakistan both tested nuclear weapons in the late 1990s. Their lack of further testing is likely because it would encourage reciprocal testing from the other country, which would allow them to advance in their warhead development. That is, as long as India doesn't test, Pakistan won't test, and vice versa, and if so, then their warheads are locked in their current state of development. If one tests, then the other will, and both will get improved warheads.
North Korea has tested as recently as 2017. They seem to be content with their current state of warhead development, or at least want to seem that way.
Whether any of these countries will stop testing indefinitely remains to be seen. The post-Cold War calculations of "freezing" warheads have been called into question many times since, including very recently, and the US is fielding a new nuclear warhead in the near future for the first time in decades (the W-93). Whether the similarity between the W-93 and previous warhead designs, and confidence in US computer models, will allow the military and political forces to avoid testing is unclear to me. Similarly if India/Pakistan decided to test again tomorrow it would only be a mild surprise as opposed to something truly shocking — it would mean that something in their internal calculus about these things had changed.
As the CTBT is not in force, the only prohibition against testing is international opinion, fears of triggering further responses from enemies (e.g., their own testing), and the norm of non-testing. None of these are by any means iron-clad reassurances. Both the US and Russia (and probably China) have considerable means of simulating and diagnosing nuclear warhead issues short of testing that produces a nuclear yield (e.g., "zero-yield" testing, computer simulations, and so on), which helps reinforce the lack of testing, but may eventually reach a political breaking point (where the uncertainties compounding in an arsenal divorced from testing finally get to the point that someone orders a test).
It should be noted that freezing warhead development only froze one aspect of the arms race. The delivery vehicles — rockets, bombers, subs, etc. — were still tested are still be developed and improved to the present day. In many ways the freezing of warheads has not changed very much, at least between the US and Russia, because both nations had already amassed such testing data and warhead design information that they could field any kind of warhead they could imagine using on new delivery vehicles. Nuclear testing is one aspect of the arms race, but hardly the only one, and the shuttering of the test sites may have done some good, but it also obscured the fact that the nuclear weapons did not go away and in fact are still a major part of international tensions.
An excellent resource on how the US saw the end of testing is Hugh Gusterson's Nuclear Rites, which is an ethnography of the Lawrence Livermore National Laboratory in the 1990s, and shows how testing was seen through the eyes of the weapon designers, and what its end meant to them.