In Italian the double negation is generally used with a negative meaning, like in the following examples
Non conosco nessuno
Non guardo mai la televisione
Non posso farci niente
The way you can think about this is to consider the first negation as not having effect on anything else apart from the verb.
With this "rule", the non only serve the purpose of turning the verb into its negative form, but it doesn't affect the rest of the sentence.
This is also coherent with some other examples in which two negations on verbs make the sentence a positive one
Non credo di non essere capace
Non dico che non sia appropriato
In both sentences the negation is attached to the verb, and two negated verbs turn the sentence into a positive one. It's worth noting, though, that
Non credo di non essere capace
and
Credo di essere capace
although both expressing a positive sense, are not interchangeable, the former expressing a higher degree of doubt about the subject's abilities.
Finally, as an addition, Italian is not the only language making an extensive use of double negations. Spanish is another notable example:
No conozco nadie
No puedo hacer nada