The answer is a little boring, as it boils down to being more like Western written languages through efforts to develop. I'll go over a couple larger languages for the sake of illustration, but that's going to be the common theme here.

Chinese

Written Chinese lacked punctuation until the first quarter of the 20th century. Hu Shih and Zhou Zuoren submitted the first serious proposal, and from there, "Western" style punctuation came into play. This was all part of a larger push in the period that ultimately lead to the literary form of Chinese (文言文) being replaced with a colloquial style, in the fashion of European writing, and also brought about things like changes in the pronominal system in writing, specifically adoption of a feminine third-person-singular character tā 她, though in speech there's no such gender distinction. All of this was a result of the May 4 Movement, which is a huge topic on its own and so I won't get too into it, but it was an active push to modernize society and get away from the imperial history. Around this time a great many things were happening in linguistic spheres to shift the way written language was handled, as well as things like the standardization of spoken language. While some punctuation specific to Chinese still exists and is used, in general, with a shift from vertical to horizontal writing, punctuation has become more or less the same as in English. Note that a full stop <.> is <。>, and some other things like Chinese-specific brackets exist, but nothing would seem shockingly out of place to an English reader.

As an aside, one reason punctuation wasn't really a part of written Chinese earlier on is because it's just not needed. A number of "words" which would be pronounced out loud in speech served the purpose of punctuation marks, such as ending statements with 也 ye, or in modern Mandarin how yes/no questions end in 嗎 ma, which would make it perfectly clear how a sentence was to be parsed.

In the Classical Chinese mentioned above, you also had pronounced particles that made it clear when a sentence was over, such as ye 也 at the end of declarative sentences. It had other roles, depending on the author and period, such as serving as a topic marker, but we're keeping it a little simple here.

Arabic

Punctuation was also not always present in Arabic. In fact, like Mandarin, Arabic also has a little bit of this kind of particle for questions. For example, yes-no questions start with /hal/ هَـلْ which, when you see it, clearly marks the beginning of a sentence, and also you know it's a question even if there is no ؟ mark. Likewise things like the vocative "ya" يا that occurs before names help in parsing clauses. This comes before a name when you're speaking to someone, so "Thank you, Ahmed" would be شكرا يا احمد shukran ya Ahmed, and from it's placement you'd know where phrases were beginning and ending.

There's actually a great paper by Dana Awad titled The Evolution of Arabic Writing Due to European Influence: The case of punctuation which goes over this topic nicely. I recommend just reading through it since it's not a difficult read, but a quick quotation in the mean time:

The importance of [punctuating] non-religious texts was considered only in the late nineteenth century. The writer Zaynab Fawwāzwas inspired by French punctuation and suggested including European punctuation into Arabic texts in an article published in the Egyptian magazine al-Fatāin 1893. Following her suggestion, Ḥasan aṭ-Ṭuwayrānī, the editor in chief of an-Nīl newspaper at the time, tried in vain to “create” signs synonymous to those used in French writings in order to avoid“borrowing” from foreign signs.

Emphasis is my own. It was, again, an active effort to bring the writing system more in line with European texts.

Other languages

Ok I admit all my examples are from languages I have a high degree of familiarity with, and it's possible that there's an indigenous African language which has similar punctuation to English just by coincidence, and I happen to know nothing about it. Still, on to scripts like Burmese and Devanagari, both of which are derived from the Brahmic script.

This is a good case where we don't have English-style punctuation. In Burmese, the full stop is two bars, looking like ||, and a comma is one bar, |. Hindi also uses the bar, but only one is needed for a full stop |. However Hindi has also adopted a bunch of other punctuation marks you're familiar with, but Burmese has not done this. Thai, which has a related script, also uses many modern marks that you'd be familiar with, but has nothing for a full stop. You know where a sentence ends because of the grammar. In Burmese, actually, you can also have really long sentences that are themselves made up of smaller sentences that could well stand along, but are joined by a number of linking particles, which in turn helps the reader/listener parse what is being said to them without confusion.

You can probably imagine English without any punctuation marks it would still be readable you would have grown up used to it and you wouldn't really be confused where each sentence ends it would be clear enough from the grammar and word order as it is in many other languages.

I mentioned 也 as a topic marker above. I should note even modern colloquial Burmese also has similar markers. For example ka က which is used as a topic marker and can occur in sentences which otherwise lack a verb (i.e. we can translate it as "is" or "to be"). These kinds of subject/object/topic/focus markers help parse sentences without any punctuation because it tells you which part of the sentence is doing what role, and in a language like Burmese which, like nearly all of Sino-Tibetan other than Sinitic and Karen, is a verb-final language, there's no question where a sentence begins and ends.

New scripts

There are a number of languages with no writing system for which systems are currently being developed. I've consulted with a number of people who are doing this sort of thing for their own languages. Punctuation always comes up as something to discuss. And in some cases, people actively reject European punctuation in favour of creating their own to match the system they've developed. For others, the creators don't particularly care, and adopt the same punctuation we'd use in English. It's often an active decision though when people develop their own, and really comes down to the personal decision of the primary creator.

Anyway,

back to the original question, in most cases where we have modern punctuation looking a lot like that found in European languages, it's generally a result of active efforts to "modernize" the script, and it's easy enough to adopt what is already globally common. Same reason the numbers we use in English are found around the world, even in languages where they have their own numbers.

Finally, in case it's no obvious, we don't have punctuation in speech. And while we have varying degrees of pauses which are now easily encoded through different conventions in writing, this was not always the case in English and still we can get by. So it's not imperative that punctuation exist. We can still make do and keep up with what is being communicated. There is a lot of redundancy in language and communication, and in part this is to help filter out the noise and keep things clear.