To the extent that this is actually a popular understanding, it's pretty much wrong in every imaginable way, including the "we have guns and you don't".

A really abbreviated sketch of an answer is this:

a) During the initial period of European engagement with sub-Saharan Africa in the early modern period, Europeans rarely engaged in the kind of colonial conquest that they pursued in the Americas. Most European presence was confined to coastal trading stations. But there were exceptions: the Portuguese had an expanded military presence in parts of what is now Mozambique, in what is now Angola, and also along the coast and far western portion of present-day Congo. Another exception: the Dutch East India Company outpost in Cape Town produced a process much like the European colonial settlement of North America--successive waves of discharged company employees were given leave to take land in the Western Cape, and were sometimes defended by Company military power against local people. In the early 1800s, the British took over the Cape and began a more active, if spasmodic, process of imperial conquest of African societies, beginning with the Xhosa chiefdoms of the Eastern Cape.

b) Despite the term "Scramble for Africa" commonly applied to the late 19th Century, Europeans were involved in more active military operations in parts of sub-Saharan Africa at a much earlier date, if in a fragmentary and inconsistent fashion. A good example might be the advance of French colonial authority up the Senegal River from the port community of St. Louis. An ambitious French official named Louis Faidherbe pushed the process along, sometimes against the wishes of superiors back in France. To simplify it to a bit of a caricature, Faidherbe would essentially claim that he had to send the relatively limited forces at his regular command, along with those of African allies, a bit further inland in order to protect existing French settlements or interests against a nearby hostile chiefdom or state. Having taken punitive action, he would write home to say that he had been reluctantly forced to claim new territory on behalf of the French nation as a buffer zone. And repeat.

A more dramatic example was General Gordon in the Sudan. Gordon had been allowed to serve the Khedive of Egypt as a sort of advisor and had come to believe in the importance of extending British and French authority across a wider region of Northeast Africa. While back in England, he heard that a new Islamic leader was threatening to attack Khartoum (now in present-day Sudan) and accepted an assignment to evacuate British-loyal local people. He decided instead to use his military resources to fight the Islamic leader. The British government of the time wanted to avoid further expansion of imperial loyalty, but the refusal to send aid right away to help Gordon actually led to the fall of that government and the dispatch of a relief force (which arrived too late)--which in turn led to the Sudan becoming a colonial possession.

c) Even at the height of imperial conquest from 1880-1910, the process could be pretty uneven. You had "company militias" like the one organized by Cecil Rhodes to claim what is now Zimbabwe, Zambia and Malawi that marched forward, signed treaties that simply guaranteed their right of residence and to work some land or mines, but which then met with popular resistance or rebellion by local peoples, defeated those rebellions, and ended up claiming imperial sovereignty. You had the Germans shelling the coast of what is now Tanzania until they were acknowledged to be imperial rulers. You had direct, bloody conquest by land armies followed by formal treaties of surrender, as in the case of the last Anglo-Asante War in what is now Ghana. And in many cases you simply had a European power claiming territorial power over a large area but not really exerting it in any meaningfully real way for another twenty or thirty years--parts of French West Africa didn't really functionally fall under administrative authority for about that long, for example.

In many territories, imperial control was met with armed resistance--there were numerous small and large uprisings and rebellions against European authority into the 1920s in many regions, and the threat of such rebellions never really died down throughout the imperial period. Sometimes these rebellions were met with extraordinary violence on the part of European administrations: the German government committed genocide against the Herero people of what is now Namibia, for example.

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d) African rulers and elites sometimes acquiesced either implicitly or explicitly to colonial rule, and in a few cases, could be seen to be requesting the establishment of imperial authority. That was generally a strategic move intended to get the best possible terms for what seemed like inevitable capitulation. A good example of this would be the Bechuanaland Proectorate (now Botswana), where a delegation of rulers went to London to ask to be brought into the British Empire as a protectorate in order to forestall being put under South African (and thus Afrikaner) authority. It wasn't that they were eager to be colonial subjects; it was in their judgment the best of a bad situation. Africans did sometimes also work with an advancing European force in order to spite nearby rivals or enemies, a choice which they almost always came to regret quite quickly--in many cases, societies that had been rivalrous in the 18th and 19th Century quickly cooperated together in rebellions or uprisings only a decade or so after conquest.

e) Force asymmetry in the form of guns did matter, but in many cases, African societies did have guns in some considerable amounts. It's just that the guns they had were not able to match the repeating rifles, machine guns and artillery of European industrial militaries. Other advantages that had previously created relative parity between some African and European militaries also disappeared at the same time--quinine provided Europeans protection against malaria, rapid communications and fast shipping led to reinforcements being available quickly, etc.

You can definitely question, however, whether it was really a conquest in the fullest sense, at least in those first few decades. European imperial powers were able to violently extract some resources "at the point of a gun"--most infamously rubber in Congo--but in many places, rural communities more or less went about their business as if European conquest had never really happened. If an administrator showed up to collect taxes, people just left and hid; if a chief was ordered to provide young men for mines or road-building, he might ignore the request or send one or two token men that he wanted to punish. It wasn't really until around World War I that those administrations began to have real power on the ground. Describing how that happened and why would be another very very long answer, much longer still than this one.

Thomas Pakenham's Scramble for Africa is a decent if sometimes plodding narrative history of the actual military engagements involved in European imperialism in Africa. If you want a good case study of how it went in a worst-case example, Adam Hochschild's King Leopold's Ghost is a visceral and gut-churning place to start. Tamara Giles-Vernick's Cutting the Vines of the Past is a good up-close examination of how slow the actual advance of real administrative power was, in this case French authority in what is now the Central African Republic.

I am studying the story of today's Zimbabwe and Zambia, so I know rather that story. In general, from the 15th century Portuguese merchants appeared on the shores of Africa, as they were looking for an alternative sea route to India. They created the first european settlements for the sole purpose of trade. They fortified these harbors as well, and they did not intrude deeper into the land. What Portuguese were looking for in Africa as trading items, they got it from native traders. Also, most of the rivers of West and East Africa have many cataracts, which hindered river navigation, and the dense forests of Central Africa hid its resources. Therefore, Africa was not particularly interesting for European colonizers.

As far as I know this was the characteristics of colonization up until the 19th century. During the 19th century the unified Germany was on the rise and she tried to establish colonies like Britain, Spain and the rival France. The hunger for colonies of the newly emerged nation states fueled geographical expeditons and disoveries. As Africa was mainly unexploited, this continent became rapidly the target of these powers. Because of this rivalry any patch of unclaimed land was potentially a future colony of someone, so every colonial power tried to arrive and claim the lands first. This kind of colonization differed from the earlier practice, as it was more violent and was partially fueled by racist supermacy theories as well. But indeed, what was the actual mechanism?

Hopefully someone share the french, late-portuguese and german practice, I can only write what in today's Zambia and Zimbabwe happened. In South Africa (Cape colony) Britain got a foothold by the Napoleonic wars (Convention of London, 1814). Here they extended the colony by gaining territories from the Boers (former Holland settlers). Boers simply acquired lands from natives by moving there and start farming. They had little conflicts with bushmans, and more conflicts with zulus. This was a mixture of migration and sort of wars. Around 1880 the Boer free states discovered huge deposits of gold and diamond on their territories. In multiple wars Britain gained this land. This is where the actual mechanism is to be found: the colonial powers targeted mineral deposits. The concept of land claiming existed indeed, but in practice it was subordered to economic and diplomatic considerations.

Paralell to the Boer wars, a group of British investors (led by Cecil Rhodes) tried to get mining concessions in the Ndebele/Matabele kingdom. This was a tribal kingdom north to the Boer states. The ruler called Lobengula, his people were heardsmen. Interestingly, Ndebeles weren't native in this region neither, they were pushed northwards by the Boer expansion. The extention of this kingdom was huge, it streched from one colonial territory to the other: to the west and east were Portuguese territories, to the north there was Belgian Kongo and German Tanzania. German, Portuguese, Boer and British businessmen tried to get a mining concession from Lobengula, so they set up kind of embassies in Lobengula's village. The king noticed that any colonial power could easily put him aside, but as soon as one would do that, all the others would create a coalition against that one. Therefore Lobengula tried not to favor any of these lobbists, or if so, then he gave just little things such as allowing to enter some missonaries. By these little things he lowered the frustration of the lobbists. Yet, at one point it was no longer sustainable this kind of policy, and finally he granted mining concessions to the British, although a very limited one: only for one mine with a few workers outside any Ndebele settlements. He choose the British because they seemd to be the most powerful so he thought the Germans and Portuguese would accept his decision. That part of his calculations was correct indeed, but the British delegation acted not according his plan.

The activity of the British delegation was characterised by Cecil Rhodes who wanted that concession very much, and earlier he even tried to push the British Colonial Office to decide to invade Lobengula's kingdom. But neither the Crown nor the Government wanted further expansion, so they made an agreement: if Rhodes gain a concession from Lobengula, the Crown will issue a royal charter (a permission) for Rhodes' private company (British South Africa Company, BSAC) to colonize the Ndebele kingdom in the name of Britain. Despite the concession was not undoubtedly authentic, the Colonial Office accepted it, and Rhodes could start the mining operations. He established a private security unit to protect his miners, and started to march into the kingdom. As Rhodes sent much more miners as allowed, Lobengula sent out his army. An uneven fight had begun, and Lobengula lost. A year after he died and his successors were also defeated by the BSAC private army. By the concession Britain could prove for the international audience that they fought a rightful war, so neither Portugal nor Germany did help out Lobengula.

Based on this story I would state that colonial powers were more cautious than just invade the natives described in your question, but mostly because of other rival colonial powers. Native tribes were seldom such unified as Lobengula's land, but surely they resisted the colonizers, they become colonies not willingly. But it is also true, that the natives had other concepts of rule as the colonizers, in some cases the natives perceived the european presence as co-existence: the europeans do mining, the natives care with the herd, and as long as the colonizers didn't want introduce taxation and compulsory education, natives didn't think they are someone's colony.

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