During the age of Metternich, Habsburg Austria was a staunchly conservative patchwork of feudal territories. How is it that by the 1800s, the country had numerous liberal governments, created a capitalist economy and was the founding center of liberal Austrian Economics?
The largest group of liberals (though not the only one) in Cisleithania were the so-called German liberals, which had their roots in the 1848 revolutionaries, who were post-1848 forced into underground and started resurfacing in the 1860s after a period of neoabsolutism and beginning of the constitutional era, when they formed the "Deutsche Verfassungspartei" ("German constitutional party"; the name of the party/political club then variously changed over time). They were the strongest party in the late 1860s and 1870s (eg. they got 218 out of 353 seats in the Reichsrat in 1873, the lower chamber of the Cisleithanian parliament, thus gaining a comfortable majority) , though they started gradually losing influence and relevance during the Taaffe's conservative "Iron Ring" government between 1879 and 1893 when they were in opposition, and definitively imploded in 1895, when they lost their relevance primarily to radical nationalist and socialist parties. It is also worth noting that the liberalization of the economy actually pre-dated the rise fo the liberal movement in late 1860s; significant steps were already made during the neoabsolutism of Alexander Bach, who by the way was originally a liberal during 1848 before he switched to a proponent of neoabsolutism.
Now, why were the German liberals so strong? First off, as I mentioned before they originally consisted mostly of the relatively well known 1848 revolutionaries, who consisted mostly of the powerful and wealthy German-speaking bourgeoisie, whose interest they represented, which is relevant due to the fact that the census suffrage skewed the elections in their favour (more about that below). The constitution of 1867 (as well as the earlier February patent was also primarily seen as being "their" victory, which boosted their popularity.
Regarding the wider reform of the Habsburg Monarchy in the 1860s, you have to keep in mind that the Monarchy suffered some major losses on the battlefield (Second and Third Italian war of independence, which also meant a loss of Lombardy which was a very economically important region, and the Austro-Prussian war), diplomatic isolation and economic/administrative problems combined with growing nationalist tension throughout late 1850s and 1860s that emboldened the reformers as well as the nationalist agitation throughout the Empire thus the situation was no longer managable without some major reforms. Hence in the mind of the Emperor and his advisors, the Monarchy was on the brink of revolution and possible collapse (after all Franz Joseph lived through 1848 that almost caused the collapse of the entire Monarchy and definitely did not feel like repeating that experience anytime soon) and the way to save it laid in at the very least the Ausgleich with Hungarians and appeasing at least to a degree the domestic liberals by creating a proper constitution and parliament.
The Constitutional Party despite the name wasn't really a single party, but more of a grouping of various liberal movements with varying level of "integration" into the party itself that also changed over time, for example "Partei des verfassungstreuen Großgrundbesitzes" (could be translated as the "Party of the Great Land-owners Loyal to the Constitution"), which consisted mostly of high income people, such as liberally minded nobility, big businessmen etc., who were only loosely associated with the Constitutional Party; then you had the so-called Old Germans (or Old Liberals) and Young Germans (Young Liberals) with the former being less radical and nationalist than the latter.
Regarding their programme and allure - they were proponents of a strong centralized state (thus no concessions regarding possible federalization) and anticlericalism (by the way often invoking Theresian and Josephine reforms) or limiting censorship, they also believed that they were the chief protectors of the constitution and continued with the liberalization of the economy, they were also German nationalists - they believed that it was the German peoples that were the "Staatsvolk" ("State people") of the Habsburg Monarchy in a sense that they were indispensable to the construction and maintenance of "Austria", that German peoples were inherently liberal and that German language and culture was naturally superior to others, in particular the Slavic ones (they claimed for example that the Slavic culture, language and scientific achievements were vastly inferior to German ones, thus in their view the former was completely useless) had which they believed would be eventually assimilated into the German one, which they believed also was in the Slavic interest to do so (there was an internal split in the party regarding this possibility of assimilation of non-German groups later down the line during their time in opposition), this was also partly influenced by their belief that modern state should be centralized and monolingual, again invoking the spirit of Josephinism. Eduard Herbst, a prominent German Liberal from Bohemia, in 1866 somewhat nicely summed this programme into his motto: "Austria, German nationality and Freedom" (ie. centralism, German nationalism and liberalism).
Now, as I mentioned their programme was mostly alluring to the German-speaking bourgeoisie, that was wealthy enough to fully take advantage of the liberalization of the economy, and which was to a significant degree influenced by German nationalism, so the German-centric/chauvinist programme appealed to them. It also represented a way for them to displace the nobility as the main political elite. The significance of this social class was further increased by the census suffrage - the votes had different weight depending on the amount of taxes an individual paid^1. Since they were relatively wealthy, their votes carried significant weight and with their votes the Constitutional party held major influence in the Reichsrat. Their dominance was also further exacerbated by the fact that the "arch-enemy" of the German Liberals, the Czechs, who were also the second largest nationality after Germans in Cisleithania, were enacting a policy of passive resistance, when they until 1878 refused to send deputies to the Reichsrat, hence the German liberals also faced little opposition.
However, as was mentioned, the German Liberal camp started suffering significant setbacks in 1879. Their programme based so heavily on German nationalism,centralism and anticlericalism, more or less caused that they had no other allies in the Reichsrat other than themselves, as because of the first two factors they couldn't really find much support in the representatives of other nations (most notably the Czechs; though it is worth noting that the Poles of Galicia were occasional allies with the German Liberals) and because of the last factor they had no backing of the church and more conservative noblemen.
This started becoming very problematic not only as the party was becoming more and more split on certain issues, however also when they needed allies against the government. The Liberals under Eduar Herbst were opposed to the government policy of expansion into the Balkans, or more precisely to the occupation of Bosnia in 1878 (which was also the year the Czechs returned to Reichsrat); however more or less everyone except them supported the government position and they found themselves alone and isolated. Moreover the return of the Czechs to the Reichsrat as well as the conservatives won a few swing votes in the Great Land-owner curia meant that in the 1879 elections the Taafe's "Iron Ring" consisting of conservative Hohenwart's political club (that also consisted of the Slovene representatives), Czechs, and Poles formed a majority in the Reichsrat up until 1893.
Taaffe started implementing concessions to the other nationalities, particulary regarding language, which infuriated the centralist and German-centric German Liberals. During their time in the opposition they became more increasingly split and more nationalist, with their older, more moderate members losing influence to the more nationalist-minded ones. Also the gradual changes to the suffrage law meant that they could no longer fully rely on votes from the bourgeoisie and started losing influence as a whole to the increasingly more powerful socialist and nationalist movements. In 1895, the party finally imploded with the so-called "Fortschrittsklub" (Progressive Club) splitting off and creating the independent "Deutsche Fortschrittspartei" (German Progressive Party) that was aligned along the Young German lines; and other members of the former constitutional party joining either socialist or nationalist parties.
Source: KWAN, Jonathan, Liberalism and the Habsburg Monarchy, 1861-1895; 2013
^1 To elaborate how the suffrage system worked: you had to pay a certain amount in direct taxes (the minimal amount was being gradually lowered) to even get a right to vote, thus the vast majority of the working class or the poorer part of the population had no right to vote. I you did pay enough taxes you belonged into one of the electoral curias and the weight of your vote was dependend on which curia you were a part of:
Later on the general curia was added by Badeni in 1896 which while holding the least weight allowed every male citizen not qualified to the other four curias to vote; the curia system was finally abolished in 1907, when every vote became equal.