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Unlike the so-called "Age of Exploration," this period doesn’t have a codified name, in part because I don’t think that it has as large a place in our popular narrative of "history of Western civilization" the way that earlier explorers like Columbus, Magellan, and others do. However, that doesn’t make that we can’t talk about it. My answer here is going to focus on Britain and the British Empire specifically, both because that’s where my area of expertise lies and because they were all over the globe and had an intense interest in exploration and the "adventures" of these explorers and the knowledge they gained permeated British popular culture. However, the British weren’t alone in this endeavor, and French, Americans, and other nationalities took part. This topic is just too broad for me to go into detail in all times and in all places, so I’m going to try and zero in on a few specific examples that I think are illuminating and also talk some about the popular culture aspect.

As you hinted at when you said that India was at the height of colonial rule, all of this exploration was a facet of imperialism. While the British and other Europeans set out to conquer the globe, they wanted information about the places they were colonizing—clear definition of borders, knowledge of the local geography, and a catalogue of potentially-exploitable resources.

However, this white European incursion (particularly into the African interior) wasn’t just driven by practical considerations that would assist the running of empires. The Victorians in particular had a huge appetite for knowledge and a desire to describe and classify the world.

I’m going to pause here with a disclaimer because this answer is going to be discussing "knowledge" and "science" a lot, and I want to be clear that I’m talking about white European knowledge and ways of knowing. Obviously, the indigenous people who lived in the places being systematically explored for the first time by white Europeans knew their own landscapes and environments, and European exploration pretty much always relied on local guides and porters.

That said, this drive to increase European knowledge was an outgrowth of what the Victorians believed to be a rational, scientific mindset that would add to the huge knowledge-base of British civilization. I’m going to quote a passage from Joseph Conrad's 1899 novella Heart of Darkness, both because it’s exactly the time period you’re asking about and because I think it’s instructive:

Now when I was a little chap I had a passion for maps. I would look for hours at South America, or Africa, or Australia, and lose myself in all the glories of exploration. At that time there were many blank spaces on the earth, and when I saw one that looked particularly inviting on a map (but they all look that) I would put my finger on it and say, 'When I grow up I will go there.' The North Pole was one of these places, I remember. Well, I haven't been there yet, and shall not try now. The glamour's off. Other places were scattered about the hemispheres. I have been in some of them, and... well, we won't talk about that. But there was one yet—the biggest, the most blank, so to speak—that I had a hankering after.

The speaker, Marlow, then goes on to describe the Congo River and eventually goes there, where most of the book is set. We can see in this passage that as a young boy, Marlow wanted to fill in all the blank spaces on the map. Marlow is, of course, a fictional character created by Conrad, but Marlow wasn’t a weird child but rather emblematic of his culture in nineteenth century British—part of the British imperial project was indeed filling in the maps (preferably with red or pink, the color used to denote British imperial holdings). This is all to say that geography, scientific knowledge, and empire are all enmeshed with each other such that you can’t really untangle them. Added onto this was an ideology of "civilization," in which imperialists also believed they were spreading their superior civilization by making contact with indigenous peoples (and in some cases converting them to Christianity).

The case of Dr. David Livingstone seems instructive here. The phrase "Dr. Livingstone, I presume?" is still famous, but most people don’t know why. Livingstone was a Scottish doctor and Congregationalist missionary who spent extensive time in Africa. Livingstone represents the complicated admixture of imperialism, humanitarianism, and science that often underpinned these explorations. Livingstone began his career as a missionary intending to convert Africans to Christianity and eventually led scientific expeditions mapping the African interior. He was the first known European to see Victoria Falls and did much work mapping the Zambezi River (through much of what is today Zambia) with the backing of the British government.

These expeditions made Livingstone a hero in the eyes of the British public. He represented, to them, the best of Victorian society, a missionary with a sharp scientific mind making discoveries for the good of the nation. Livingstone used his fame to oppose the Arab slave trade in East Africa.

In 1866, Livingstone returned to Africa with the intention of locating the source of the Nile River. Keeping in mind that this was before technology like satellites and GPS that aid in mapping and that much of the interior of the African continent was still unknown to Europeans, this was a topic of debate amongst explorers and geographers. The source of the Nile had been identified by John Hanning Speke and Richard Francis Burton (other explorers who both could be subjects of their own posts entirely) as Lake Victoria, but Livingstone disagreed and set out to prove them wrong. (Speke and Burton were more or less correct, by the way, but even a quick perusal of the Wikipedia page on the source of the Nile will give you an idea of why there was so much confusion; several other rivers feed into Lake Victoria itself.) Livingstone was soon in the African interior without contact with the British, and as the years wore on without his return, the public began to worry about him.

Enter Henry Morton Stanley, a Welsh-American whom the New York Herald paid to look for Livingstone while sending dispatches about his journey. Stanley's dispatches were a sensation in the press, and he eventually found Livingstone in the town of Ujiji on Lake Tanganyika, hence "Dr. Livingstone, I presume?" (Since Livingstone was the only white guy around, it wasn’t hard to figure out who he was.) Livingstone chose to stay in Africa, continuing to search for the source of the Nile, and eventually died there. Stanley returned to tell his story and became an explorer in his own right, though his reputation is considerably darker than Livingstone's, despite some attempts to rehabilitate him by biographers such as Tim Jeal. When you help King Leopold of Belgium establish the Belgian Congo, one of the most brutal of imperial regimes, history doesn’t look kindly on you. He also had something of a reputation for personal cruelty to his African workers, which doesn’t help.