From my understanding of history, I have a rather different take. As a practical matter, it seems that a number of the expansionist policies of Tsarist Russia were continued under the Soviet Union. Among the policies which either continued or in some instances was dramatically increased in effect was the encouragement given to ethnic Russian re-settlement/colonization in Soviet Socialist Republics other than Russia. The current Russian city of Kalingrad being an example, where the site had been the capital of German East Prussia and called Konisberg with a native German population, and following WWII the native inhabitants had either fled or were deported en masse with Russian re-settling the area after the Soviet Union claimed additional territory from Germany and Poland.
The Soviets had also gone about re-claiming territory which had been held, some albeit loosely, by Tsarist Russia but lost during the Russian Civil War as various ethnic minorities re-asserted their independence. This was accomplished with a mix of annexation and occupation like was done in the Baltic States and portions of areas claimed by Finland and lost to Russia during the Winter War.
With all the above having been said or mentioned, I would consider it highly unlikely that nations in the early 1990's, which had formerly been under Soviet (Russian) rule or influence, would not automatically assume that a future resurgent Russia would not desire and likely seek to reassert control over areas that had once been under Tsarist and/or Soviet dominion. Given some of what has gone on with Georgia, Moldova and the Ukraine as well as the conflicts with Chechens in the 1990's, it would seem that there were some within the halls of power in Russia who were going to seek to 'restore' what had been lost regardless of what NATO did.
All the above should certainly illustrate why various former Soviet or Warsaw Pact nations would be interested in an opportunity to join NATO, but it does not really answer why NATO would allow former adversaries to join. At this point I do feel the need to point out that, apart from the former East Germany/GDR becoming part of NATO in 1990 following the reunification of Germany, the next nations to join NATO, namely Poland, Hungary and the Czech Republic did not join NATO until 1999, at the end of the 1990's. Two reasons come to mind. The first being that the free and independent countries might feel a moral obligation to aid other independent nations in maintaining their sovereignty, particularly given the history of the region. The second involves a degree of self-interest, or at least the potential for it. This involves establishing a framework which would make it more difficult for a future Russia re-assert power/control over former holdings and client-states, and thereby attempt to avoid or minimize the scope of future conflicts. Looking back at history, consider how WWII might have been different, or perhaps never even occurred if then-Czechoslovakia had been stronger and/or had a sufficiently strong mutual defence pact which could have prevented the annexation of the Sudetenland.
While Russia has been able to annex the Crimean and de facto occupy portions of the Ukraine, Moldova, and Georgia, it has not been able to do something like that with the Baltic states, who are NATO members. As for whether or not that makes sense, it does raise the question of whether it is better to cater to one nation which is a major power, or a number of smaller nations who are concerned about the major power.