Why do some languages partition 'to know' into 2 or more verbs?

As always, why questions are always bad questions to ask in linguistics if you're asking about some putative motivations of historical speakers. We don't have the data (with some rare exceptions like "freedom fries"). At best you can trace the historical developments within a language and note the differences between languages.

This type of difference where one language uses several words (or concepts) to apportion reality differently is so common that any individual instance is nothing more than an idle curiosity. It is one of the foundational definitions of language itself that goes back to de Saussure who talked about the net of language being cast on the amorphous sea of reality.

One of the languages that differs from English in this respect is Czech which has three verbs (Russian for instance, only has 2).

vědět - to know in general (intransitive - except it can take 'it' and 'something' as direct objects as a in 'I know it' or 'I know nothing', I know that water boils at 100 degrees C.)

znát - to know someone, some place, be cognizant of (transitive: I know Peter, I know London, I know (of) this song)

umět - to know how (I know how to swim, I know (how to play) this song)

But interestingly, the two languages meet when these verbs are nominalized. English has 'knowledge' and 'ability' and Czech has two nouns that match perfectly (or rather the two nouns derived from znát and vědět are almost perfect synonyms and the noun for ability is derived from a completely different verb).

There are also many other verbs related to knowledge - 'recognize', 'realize', 'distinguish'. While English has completely separate roots for these, in Czech, many of these are related to one of the verbs for 'to know' - 'poznat', 'uvědomit si', 'rozeznat'.

How did this happen? Or why is Czech different from a more related language like Russian? There is no sensible way to answer this. However, internalizing these differences can give you a better insight into how language works.