In late July and early August 1968 a Czechoslovak delegation led by Alexander Dubček met with Soviet and other Warsaw Pact delegations to negotiate over the Prague Spring reforms. The Warsaw Pact leaders felt that any liberalization in Czechoslovakia could threaten the military and strategic position of the Pact against the West in the Cold War. As I understand, this was their primary concern. So why did Brezhnev then compromise and willingly weaken the Pact's strategic position by withdrawing Soviet troops from Czechoslovakia? That seems to me like the last thing he should have compromised on, even if he believed it would only be temporary, and it seems like after that the invasion was inevitable. According to Wikipedia, "The KSČ delegates reaffirmed their loyalty to the Warsaw Pact" during this meeting as well. How does that mesh with asking Soviet troops to withdraw? Should Brezhnev have not simply refused to withdraw troops and instead compromised in another area? It seems like he wasn't negotiating from an especially weak position. Especially considering that there was still a substantial anti-reformist wing of the KSČ.
Please forgive me if I did some stupid mistake in there, some of the sources I have read through may be wrong and I didn't realize it.
In short: He did not withdraw.
During the exercise Šumava earlier that year (June 1968), there were many soviet troops which withdrew to the borders and were prepaired for an order to be given for them to cross the border back, but there are cases of Soviet military units, specially the communication units, that were delaying their departure from Czechoslovakia with fabricated reasons (from leaked information to the Czechoslovak HQ, some units were given orders to not withdraw until September 1968, officially the last units withdrew on 3rd August (that is 20 days after all the other forces), but there were still some units that remained in Czechoslovakia till 20th August).
From sources I found, there were earlier attempts to cross the border, some as soon as May, which were called off for reasons (the May attempt was called off only because Czechoslovaks were coincidentally training with a few tanks near the place where border crossing was to occur and that in turn pushed the generals to reconsider).
At some point in late July (that is between 27th and 30th, hard to say exact date), preparations were concluded for the Warsaw Pact forces to enter Czechoslovakia, which paved a way to the Čierna nad Tisou negotiations I assume you are referring to. There are cases of some stationed soviet officers and political workers in Czechoslovakia giving hints to some higher up Czechoslovak officials, that the invasion is imminent.
Although the Čierna nad Tisou meeting didn't really salvage Czechoslovakia, it gave the Soviets enough confidence to slightly delay the invasion, because they didn't percieve any imminent danger, in order to create a political backing to it (+ there were some coordination issues, especially with East Germans, which later caused them to cross the border only to disarm local border guards and take some smaller towns near it, but didn't advance further). Not only was the operation brought up to the Politbyro and approved on 17th August, but it also allowed the Soviets to search for some collaborators.
Those collaborators in early August then written a series of letters inviting the soviets, which were later used as a justification (for among those collaborators were for example Indra, Biľak or Kolder). Even though this justification didn't work in the end and the people that signed those letters were despised by the population, there was at least a slight thing the soviets could point at to show their own citizens, that "true" Czechoslovak communists were worried about the reformations happening (the fact that this anti-reformist wing almost started a coup in December 1967 and that a prominent anti-reformist general Šejna fled to the west with top secret documents was emitted for the convenience).
And no, after the invasion, the forces didn't withdraw. Yes, some forces withdrew because they were not needed anymore, stationing in a foreign country is different to occupation, but many Soviet forces, 5 divisions and many more brigades (including rocket forces), were left stationed in the country as the Central Group of Forces, many of which were withdrawn only after the Velvet Revolution in early 90's.
The fact, that some forces were withdrawn can be justified by the fact, that Soviets created multiple nuclear stockpiles in Brdy military area, ready for the strike.
[Edit] Typo in the city name: Čierna nad Tisou