Although the Grand Duchy of Lithuania was a massive state in the Medieval and Early Modern eras, it doesn't have an unbroken connection to the modern nation-state of Lithuania, nor is Lithuania the only modern state that claims the Grand Duchy as its antecedent.

To do a speedrun of history of the Grand Duchy:

The Grand Duchy formed in the 13th century as a state governing some (not all) of the Baltic tribes within modern-day Lithuania. If you want a brief discussion of where the term "Grand Duke" comes from, you can check out this comment I wrote on that topic. Much of this state formation was in response to/having to deal with continuous expeditions and raids from the Livonian Order and the Teutonic Knights, as the Baltic tribes were pagan.

The Grand Duchy expanded like crazy to the east and south under the Gediminid Grand Dukes, who took advantage of the power vacuum left after the Mongol conquest of the Kievan Rus' principalities, and then the breakup of the Mongol Empire. Already by this point the majority of people in the Grand Duchy were Eastern Slavic speakers and Eastern Orthodox Christians, despite the ruling class being of Baltic origin and pagan. The court used Slavonic as a working language.

In 1386 the Gediminid Jogaila agrees to the Union of Krewo: he is baptised a Catholic, marries the 12 year old Queen Jadwiga of Poland, and becomes Władysław II Jagiełło, founding the Jagiellonian Dynasty. The Kingdom of Poland and the Grand Duchy of Lithuania are in a personal union (ie, they have the same monarch. Usually). The Lithuanian ruling class converts to Catholicism in 1387, and the process of converting the pagan Baltic communities begins. On top of this the ruling class of the Grand Duchy becomes Polonized, ie Polish language and culture spread and basically become the high culture of the Grand Duchy.

Meanwhile Lithuania fights a series of wars with the expanding Grand Duchy of Moscow in the 16th century. Most of these wars come off with Lithuania losing eastern territory, most notably losing about a third of its territory (parts of modern-day west Russia and east Ukraine) in 1503, as can be seen in light yellow here.

Eventually in 1569 the last Jagiellonian King of Poland/Grand Duke of Lithuania unites the two realms in the Union of Lublin, which establishes the Polish-Lithuanian Commonwealth. As part of the Union agreement, much of the Grand Duchy's southern territories get transferred to the Kingdom of Poland. Basically by the early 17th century, the Grand Duchy of Lithuania was mostly the territories of modern Lithuania, modern Belarus, and some bits of western Russia. Lithuania would lose most of those Russian bits (basically, Smolensk) in the 1667 Truce of Androsovo.

Anyway, the Grand Duchy otherwise continued on as a very multinational, multireligious, multilingual entity in the greater Polish-Lithuanian Commonwealth, with an elite that was mostly Polish in culture, a peasantry that was mostly Orthodox and Slavic, and with a Baltic corner that was Catholic and "Lithuanian" speaking (I put that in quotes because there wasn't really a standardized Lithuanian language yet, and people were basically speaking local dialects). The earliest surviving written samples of the local language date to the 16th century, but there was low overall literacy, especially among these Lithuanian speakers.

ETA: I'm rushing through but I should make mention of other religious and linguistic communities, especially the Muslim Lipka Tatars and the Ashkenazi Jews, both of which communities were notable in Poland and Lithuania.

The 1772, 1793 and 1795 Partitions of Poland very rapidly swallow up the Polish-Lithuanian Commonwealth. Most of the Grand Duchy of Lithuania goes to the Russian Empire. It ceases to exist as a separate entity, although interestingly the tsars keep "Duke of Lithuania" in their formal full titles through to 1917. Ultimately the former Grand Duchy was split between a number of Governates, notably Kovno (Kaunas), Vilno (Vilnius), Minsk and Grodno, with a chunk of the former Grand Duchy being part of the Kingdom of Poland, which was a Napoleonic-era creation.

Anyway, to fast-forward a bit: Lithuania, like many other Eastern European nations, experiences a National Revival in the late 19th century, in no small part because of the failure of Polish rebellions against Russia in 1863 and subsequent Russification campaigns under Alexander II. This begins to pick up steam in the 1880s under such figures as Jonas Basanavičius and Jonas Šliūpas, the publication of Lithuanian newspapers such as Aušra and Varpas, the standardization of a Lithuanian language and orthography under such figures as Jonas Jablonskas, and the development of national Romantic high culture under such figures as Mikalojus Konstantinas Čiurlionis. This began to see "Lithuania" in terms of a national identity rather than in terms of a geographic or historic identity. And this is no small task because there were competing national ideas that saw Lithuania in the latter terms: Polish national figures very much saw the (Polonised) Grand Duchy as a Polish construct: the Polish national epic is Pan Tadeusz by Adam Mickiewicz was from the former Grand Duchy of Lithuania, based his epic poem there, spoke Polish, and was from a part that is in modern-day Belarus. More than a few prominent Polish national figures saw Lithuania as a once-and-future integral part of Poland, and President Józef Piłsudski hoped and pushed for a renewed Polish-Lithuanian federation of some kind.

However, basically the modern Lithuanian national identity saw itself as something linguistically and culturally separate from its neighbors, and was extremely worried about the possibility of assimilation by Poland or Russia. Given the politics of the 1918-1921 era, this mostly worked out in favor of an independent Lithuania for ethnically-Lithuanian territories. Germany occupied the region in 1915 and (much like in Poland), encouraged a local administration to eventually seek independence from Russia. A State Council of Lithuania was convened (Basanavičius was one of the twenty members), and it declared independence in February 1918. Although the new State of Lithuania was considered by the Council to be a successor to the Grand Duchy, it was first and foremost a state for ethnic Lithuanians, not for the entire former Grand Duchy. Germany tried to make it a monarchy under a German Duke before giving up (and giving up on Eastern Europe in general) in November 1918. The State Council did get some limited aid from Germany, and used this to fight a crazy-quilt series of wars against the Bolsheviks, Poles, and a mixed army of Germans and White Russians (who were all fighting each other as well). The end result was that Lithuania established itself as an internationally recognized state with a capital in Kaunas, and a large territorial dispute with Poland, which effectively had taken control of the Wilno/Vilno/Vilnius "Central Lithuania" territory in the 1920s, but which Lithuania claimed as its historic capital (the Bolsheviks had given it to Lithuania in the treaty they signed recognizing its independence, but most of the city and surrounding territory was not ethnically Lithuanian). Lithuania eventually got most of the Vilnius area but at the cost of Soviet occupation and annexation in 1940.

So that is how we went from a gigantic Grand Duchy to a relatively small Lithuanian nation state. Although modern Lithuania considers itself the successor to the Grand Duchy, so does Poland in many respects, as does Belarus, considering that its territory and population were most of the Grand Duchy in the modern period. An example of this is how the Vytis (mounted knight, in Polish it's Pogonya, in Belarusian it's Pagonya) is the Coat of Arms of Lithuania, and was also the coat of arms for Belarus from 1991 to 1995, until Lukashenko restored Soviet-era imagery. The coats of arms come from a symbol of the Grand Duchy that dates back to the late Medieval period.

So while Lithuania shares the name with the Grand Duchy Lithuania and considers itself the Grand Duchy's successor, it's not the only modern country that traces its heritage to the Grand Duchy: Poland, Belarus, and at least in tsarist times even Russia made claims to be its descendant.