So this is probably the biggest or 2nd biggest topic of study for political scientists specializing in Japan (primarily because of the '93/'94 Electoral Reforms which resulted from this brief period of opposition government) meaning I won't be able to give a fully comprehensive answer, but I'll try to keep it concise and related to your specific questions just so the answer doesn't get too bloated. BUT I say all that just to say there is a ton more to learn on this topic! If you're interested maybe check out any of the books listed in my sources as they're actually pretty quick reads, but very information dense.

In any event, as you noted in your question the LDP 'lost' in 1993, but this loss was not as definitive as it may seem on the surface. The LDP did not 'lose the election' they simply did not win enough seats to form a government without a coalition. This is an important distinction when questioning why the LDP came back into power so quickly, because it elucidates the fact that the LDP was STILL by a wide margin the most popular political party in Japan even during those brief few months when they were out of power. The LDP held 43% of the seats in parliament even after this election and the next closest individual part held only 13% (The Japan Socialist Party, the LDP's longtime rivals).

The reason for the LDP's drop in popularity at this particular moment was multifaceted but scholars generally single out two major causes. The first is that the LDP's longtime consensus of economic prosperity began to crack with the bursting of Japan's Asset Price Bubble. Previously Japan's ever growing economy had allowed the LDP to mobilize political pork to strategically purchase votes necessary to staying in power (largely pulled from agrarian districts) while pacifying potentially hostile urban voters with overall economic success in the country (Even if these voters did not like the LDP, they could not deny that the LDP and the Bureaucrats allied to them continued to increase Japan's GDP, resulting in higher wages and standards of living for all Japanese citizens). SO when the bubble burst and the LDP could no longer pacify opponents with economic success, these hostile voters became more likely to mobilize against them.

The second major factor to the LDP's brief loss of power was widespread scandal shaking the public's faith in the LDP and their bureaucratic allies. Because of Japan's unique electoral system (Single Non-Transferable Vote, Multi-Member District) and because the LDP was so reliant on political pork to continue its uninterrupted dominance, LDP politicians were incredibly prone to accepting illegal funds from large corporations. LDP politicians needed money to buy votes and big businesses wanted government assistance from the LDP and the bureaucracy in the form of protectionism etc (This, along with the farm subsidies used to lure the agrarian vote into the LDP's camp made up Japan's so called 'steel and rice' coalition, named after Bismarck's 'Iron and Rye').

In any event, because of the inherent corruption of this system, it was only a matter of time before the alliance collapsed. A series of scandals (most notably the Recruit Scandal [an insider trading scam] which involved multiple LDP politicians and the Sagawa Kyubin corruption scandal [where Japanese politician Kanemaru Shin accepted outright bribes from Japanese construction companies which he used to purchase hundreds of pounds of gold which he hoarded under the floorboards in his home and which was later found by authorities searching the residence]) where high ranking LDP party members played major roles resulted in massive loss of prestige in the eyes of the Japanese populace. Moreover, similar scandals resulted in a total loss of confidence in the Japanese bureaucracy [which beforehand was considered the most prestigious and effective group in Japan]. Much like the political scandals these were incredibly numerous but for the sake of brevity I'll only list one of the two most extreme examples. The Ministry of Health and Welfare had allowed HIV infected blood to be put into Japanese blood banks and subsequently used in the treating of patients (specifically hemophiliacs). Many of these patients (total number of infected was somewhere between 1,000 and 2,000) died shortly thereafter. The Ministry of Health and Welfare had neglected screening procedures for the blood because some MoHW officials had received payments from Japanese companies which hoped to have their blood used in Japanese clinics rather than imported competitor product.

Also important to note is that the previously mentioned 'bubble bursting' was blamed on the Ministry of Finance. Many officials within the MoF neglected oversight procedures intended to keep banks from making dangerous loans. This lack of oversight destabilized the Japanese financial system, drastically exacerbating the eventual collapse. While the human cost was not present in this scandal, it served to further erode the peoples' confidence in the Japanese bureaucracy.

The LDP and the various bureaucratic ministries had been linked through the '55 System (the period of uninterrupted LDP rule). As such, when both of these institutions fell prey to scandals, the Japanese people lost all confidence in them and called for their removal. The various opposition parties used this moment of weakness for the LDP to rail against the party and the bureaucracy. They promised the people they would remove the corrupt LDP from power and take power away from the bureaucrats who were now seen as corrupt, incompetent, and disconnected from the peoples' well being.

Moreover, several high ranking LDP members (most notably Ozawa Ichiro nicknamed "the Destroyer" for his role in the LDP's '93 downfall among other things) saw this as an opportunity to leap off a sinking ship. The left the LDP to form small independent parties, calling for reform. So in addition to loss of face publicly and a reinvigorated opposition the LDP also had to deal with internal dissension in 1993.

However, in spite of all this, the LDP was still very much the most popular and most powerful party in Japan at this time. The opposition seized this opportunity of LDP weakness to form and unwieldy and unstable 7 party coalition which managed to form a government and keep the LDP out of power. However, the key phrase there was "unstable." The only thing the 7 parties agreed on was that they hated the LDP and that they wanted to enact electoral reform in order to prevent LDP resurgence. Unfortunately for them, none of the parties could agree on what type of new electoral system would be best for Japan (and for themselves). The small parties wanted Proportional Representation and a party list, the large parties wanted Single Member Districts, and as a result of these disagreements this coalition government got very little done. They DID manage to enact electoral reform, but could accomplish nothing else which the Japanese people saw as worthwhile, and as a result lost the peoples' confidence. Then in 1994 the coalition splintered and the LDP was able to form its own coalition government with some of the more marginalized smaller parties (namely the Komeito, Clean Government Party, a party affiliated with Soka Gakkai, a lay Buddhist group) and retake power. Moreover, they also were able to reincorporate many of the defectors who had left the party in the reform euphoria.

Things get even crazier after that, but I feel like I've already written too much so I'll just leave it there haha. Hope that was helpful.

TL:DR - Scandals removed the LDP's long held majority as voters lost faith in the LDP and their bureaucratic allies. However, the coalition which replaced it was very unstable and quickly fell apart due to infighting. This allowed the LDP (which had never even fallen below 43% of the members of parliament anyway and had remained the largest party in Japan) to form its own coalition government and thus quickly regain power.

Sources:

Japan Transformed - Frances McCall Rosenbluth and Michael F. Thies

Democracy Without Competition: Opposition Failure in a One Party Dominant State - Ethan Scheiner

The Rise and Fall of Japan's LDP - Ellis Krauss and Robert Pekkanen