Putin says that ABMs such as America is installing, disable a country's (in this case, Russia's) ability to retaliate against a blitz invasion — something increasingly likely from NATO now as NATO has extended right up to Russia's very borders — and that Russia will not allow this disabling of Russia's retaliatory forces.
He said that "NATO fends us off with vague statements that this is no threat to Russia ... that the whole project began as a preventive measure against Iran's nuclear program. Where is that program now? It doesn't exist. ... We have been saying since the early 2000s that we will have to react somehow to your moves to undermine international security. No one is listening to us."
In other words, he is saying that the West is ignoring Russia's words, and that therefore Russia will, if this continues, respond by eliminating the ABM sites before they become fully operational. To do otherwise than to eliminate any fully operational ABM system on or near Russia's borders would be to leave the Russian people vulnerable to a blitz attack by NATO, and this will not be permitted.
