The reason for banning anti-ballistic missile systems may seem a bit counter intuitive, but it is actually very straightforward. By the end of the 1960s, the United States and the Soviet Union had reached a level of parity in their ability to annihilate one another through nuclear attack (and despite JFK's misinformed warnings about a missile gap prior to becoming president, it was the Soviets who had finally caught up with the Americans). This created the foundation for mutual assured destruction ("MAD") which ironically provided a level of security--neither nation would dare attack the other, as it could not prevent itself from being destroyed in retaliation. With MAD having become a core component of Soviet-American relations, the Soviets became more willing by the end of the 1960s to curtail the arms race, and so it looked like the United States and the Soviet Union might soon begin the process of relaxing relations and enter into a period of "detente" (which just means relaxed relations).

Anti-ballistic missile systems threatened to undermine all of this. If one side could build a defense system that could truly counter the other side's nuclear arsenal, then MAD would dissolve. One nation would have the capability of destroying the other without its own complete destruction. This would increase the likelihood of an attack and would remove the security offered by MAD. On top of that, these ABM systems would be very expensive, and if one began developing an ABM system in earnest, the other would have to do the same. A new era in the arms race threatened to emerge (note that the Soviet Union had already begun deploying an ABM system around Moscow by the end of the 1960s--a bit more on that in a second).

So ABMs would be very expensive and would threaten the stability offered by MAD. They also would not work in the long run anyway. Both sides already recognized the technology required to counter ABMs: the so-called MIRVs, or "multiple independent reentry vehicles." These were single missiles at launch that would go into low earth orbit or outer space and then convert into multiple missiles upon reentry, all with different targets. All of the warheads could not be countered by a single anti-ballistic missile, and thus one side could bypass the other's ABM system simply by overwhelming it with MIRVs. So not only would ABM systems be very expensive to implement and threaten global stability, they would, in time, become completely obsolete. MIRVs would restore the superpower status quo that existed before the ABM systems had been developed and deployed--all at the cost of hundreds of billions of dollars by both sides. At best, the development of ABMs would be a complete waste of money (HUGE amounts of money). At worst, they would bring about the end of mankind.

By the late 1960s, the need for a treaty banning ABM systems was imperative, as the Soviets had already begun developing their own. The Joint Chiefs of Staff harassed the Johnson administration with bullshit warnings about the capabilities of this Soviet system (it was actually fairly rudimentary at the time), and it looked as though the United States might begin pouring more and more resources into developing its own ABM system at the soonest possible date (which, as noted above, would spur a new round of the arms race). Luckily, the Johnson administration fully understood the foolishness of this and sought a summit meeting with the Soviets to prevent this outcome. Only the Soviet invasion of Czechoslovakia in August of 1968 stalled the Johnson administration's pursuit of this meeting (no joke: the Soviets and the Americans were set to announce an arms-reduction summit meeting the morning after the Soviet invasion). Once Nixon became president in January of 1969, he immediately sought to continue the work Johnson had started, however, and official talks (not at the summit level) began in Helsinki in September of that year. Of course, there were many complications in realizing this treaty--for example, the Russians were not so willing to dismantle the work they had already done in building a system around Moscow--though the two sides finalized a treaty by 1972 (on the point mentioned here, both sides agreed to allow the other to build a basic ABM system at two locations, which was later reduced to one). Nixon signed the treaty at his historic summit meeting in Moscow in 1972.

Although this is not essential, I'd like to point out that Johnson, in his memoirs, lamented that the Soviet invasion of Czechoslovakia in 1968 stalled the treaty by at least a year or two, which he believed cost both sides billions of dollars in wasted money and risked all of humanity. I point this out because Detente is usually exclusively associated with Nixon, though Johnson passionately pursued detente even through the Soviet invasion of Czechoslovakia. I'll take this space to also admit that I am far more knowledgeable about Johnson's administration than I am Nixon's.

Sources.

My main source for anything like this is always the appropriate volumes of the Foreign Relations of the United States series, a collection of primary documents released by the State Department for each administration (it's more comprehensive than you might think). If you can't physically sit in the archives, then this is the next best thing.

It's all online these days too!

https://history.state.gov/historicaldocuments/johnson (See in particular volumes XI, XIV, and XVII).

https://history.state.gov/historicaldocuments/nixon-ford (See in particular volumes XII-XIV).

Other sources available on request.