"We know this man."
Translation:Biz bu adamı tanıyoruz.
Things qualified by a demonstrative determiner (bu, şu, o) or a possessive determiner (benim, onun, ...) are definite by default.
So "bu adam" is a definite noun phrase.
It is also the object of "tanıyoruz" -- the thing or (in this case) person that is known.
And since it is both definite and a direct object, it gets put into the accusative case. (Remember that the accusative case is used for definite direct objects. A noun phrase has to fulfil both of those definitions, in general, to get marked with the accusative.)
In the case of "bu adam", this involves adding -ı: bu adamı.
I think taniyor is used for people whereas biliyor for things you know about, much like eski and yasli both mean old but you use yasli for people and eski for objects.
Bilmek - to know Tanismak - to meet Tanimak - to know (a person)
I'm still learning though so i could be wrong!
The usage of the two verbs is not completely identical in languages that distinguish those two (someone once told me that Turks from Germany often use tanımak and bilmek as if they were equivalent to German kennen and wissen, respectively, and that this is noticeably different from Turkish as spoken in Germany in some cases), but similar enough to be helpful, I think.