It didn't end the arms race, even the nuclear arms race. We're still in the middle of a nuclear arms race, one that the weaponeers are gleefully predicting will continue to expand.

What it did was this: in the early 1990s, the US and Russia agreed, in a show of good faith and good will, to deliberately slow certain highly-visible aspects of the arms race. You can think of it as a "pausing" of it. Many existing systems were pared back (both because of expense and lack of perceived need), but a hard "core" remained in both countries. By beginning a voluntary halt in nuclear testing, warhead development essentially stopped to a crawl. However new delivery systems, in terms of missiles and submarines and bombers, continued being developed, and indeed are still being developed.

As for why Russia and the US would agree to this slowdown — ultimately it came down to expense. Developing arms is immensely costly. Russia was in a bad economic situation, and lacked the resources of the full USSR. The US itself was being run by people who argued in favor of cutting the deficit, and were happy to use the lack of perceived threat from Russia as an excuse to cut back in strategic arms.

Again, the actual weapon systems still developed. But instead of quantity they frequently stressed quality. US nuclear weapon "deadliness," in terms of being reliable and accurate, actually peaked in either the 1990s or the early 2000s, depending on what metrics you use. This is one of the reasons that many post-Cold War administrations (notably GWB) found it possible to dramatically reduce the stockpile size without compromising the strategic capabilities.

In the public mind, nuclear weapons became deemphasized, for sure. But this just meant people stopped paying attention, not that the work itself actually stopped.

There's a lot to untagle here for such a short question.

First and foremost: the arms race did not end, it just slowed down and, arguably, took on a different guise. The economic calamity in the (now former) USSR meant that the Russians were unable to allocate as much money to weapon R&D as before and in the US, the need to develop better weapons was not as immediate with the arch-rival now collapsing. But the quest for better weapon systems never stopped on either side. Take the Joint Strike Fighter program, which has its roots in the 1980s but only really took off well after the Cold War.

Secondly: the arms race was never solely about nuclear warheads. And even if, in the latter part of the Cold War reducing the stockpile became the goal rather than expanding it. The Strategic Arms Reduction Treaty was negotiated throughout the 1980s and signed just before the USSR collapsed. So if one wants to frame the arms race as a nuclear arms race, one could argue that it ended in the summer of 1991.

Thirdly (to build on the previous two points): the US didn't forget that Russia still had nuclear weapons. That is why missile defense systems were being developed. The Terminal High Altitude Area Defense, or THAAD, system began firing tests in the mid-1990s, again well after the Cold War had ended. Russia, on the other hand, is currently commissioning its new ballistic missile submarines, the Borei-class (project 955).