Putting it bluntly - there is a general consensus among both Czech and Slovak academics that the split was not viewed favourably by the general public on both sides. This is probably best proven by the opinion polling conducted by Martin and Zora Bútor, conducted in March 1993 - shortly after the split was finalized, showing that roughly a third of Slovaks welcomed the decision, while the rest viewed the split with considerable suspicion or fear. The split acceptance among the Czechs was even lower.

Since this is r/AskHistorians, I'll take the liberty to delve a little further into the history of Czechoslovakia and Slovak autonomy within the country, hopefully explaining the sentiment that eventually led to a somewhat rash decision to split the state. If you simply wish to read about the immediate causes for the breakup, skip it and go straight down.

When the breakup of Austria-Hungary was finalized in 1918, the Entente and especially the USA had guaranteed the whole country of Czechoslovakia as a unitary state. Many of the reasons for this were practical in nature - the dissident movements of both countries cooperated deeply (evidenced by the amicable relationship between Czech-born T.G. Masaryk and M.R. Štefánik, who was of Slovak descent. Both men would eventually occupy important positions in the future independent government, with Masaryk becoming the first president and Štefánik receiving the Defense department. Another reason for the union of the two nations was the complementarity of their respective historical regions - while the Czech heartland was heavily industrialized, Slovakia's countryside was mostly agricultural. In addition to the long-term economic benefit, there was also the obvious fear of irredentism in one of the surrounding countries - notably Hungary and Germany, whose populations made up a sizable portion of newly established Czechoslovakia. To keep both of their potential adversaries in check, the Czechs and Slovaks partnered up under the banner of Slavism - as the idea of inter-Slavic cooperation was still riding an all-time high sentiment among the general public. This also applied to the Czechoslovak legions, whose organization never even assumed any segregation in the first place. The result was a unitary state that made no special effort to give either of the nations any particular notion of autonomy. It's important to note here, that the majority of the most important governmental institutions and agencies were concentrated in Prague, partially contributing to a feeling of disconnect in other parts of the country.

Fast-forward to 1945. Prior to WW2, Slovak fascists succeeded in convincing Hitler to establish a separate reduced puppet state for them. Though the country was reunified after the war, Slovaks continued to maintain a previously unseen token of autonomy - the Slovak national council. It was certainly not powerless, but the asymmetrical nature of this establishment made any attempts to equalize the infrastructure pointless - (e.g., there was never any need to establish a Czech national council as a mirror at this point). This arrangement would hold on right up until the country was federalised in 1969 - at that point autonomy was equalized between the Czechs and Slovaks, with each respective council receiving the same power and jurisdiction.

This state continued up to the year 1990 - after the communist regime was toppled a year prior in a peaceful revolution, the Slovak national council pushed for a reorganization of the country and separate institutions. Ironically, this only created more issues as proper equalization was seldom conducted, with Czechs usually just maintaining the "old" unitary institution, while the Slovak national council tried to create their own national institution (to better illustrate - the Czechoslovak TV continued to operate in Prague with no plans to change, while a new Slovak TV was established in Bratislava). The rise of populism on both sides gave way to the outbreak of the so-called "Hyphen war", where authorities argued over the grammatical viability of a new official name for the country - Czecho-Slovakia, instead of the original Czechoslovakia, to symbolize the equal standing of both republics within the federation. When populist politician Vladimír Mečiar won the 1992 elections in Slovakia, the arguments escalated and eventually a joint agreement to split the country in two was agreed upon by both national councils. The parties in power effectively fulfilled their election promise, as Mečiar campaigned on the premise of "full confederation or separation" and the Czech PM Václav Klaus campaigned with the "fully unitary state or separation" memo.

To date, this decision has been viewed controversially - if either of the republics decided to leave the federation, a referendum would be triggered to decide on the matter. Given the prevailing sentiment described in the first paragraph, it is unlikely that this would have been successful. But Mečiar and Klaus portrayed the split as a complete dissolution of the federation (even though the Czech republic later continued to use the flag and generally portrays itself as a successor state), which was not a constitutionally defined option and would thus have no plebiscite requirements attached. However, prevailing public opinion still agrees today that dissolving the country without a referendum was a grave mistake. Juristic opinions also tend to agree that the split was indeed unconstitutional, because there was no constitutional provision for the federation's termination. With that being said, roughly 40-42% respondents in both countries agree today that the breakup was beneficial.

Summed up - the country split up because both elected representatives pushed for dissolution largely on grounds of potential personal gain, even though the public was opposed to such a move at that time.

Since this is a very niche topic for English publications, and the vast majority of my sources are Czech or Slovak, I didn't attach any recommended titles for further reading. Should somebody request them, however, I'll dig in to find something worthwhile and update the post.