Learn Italian: Non penso a niente. - I am not thinking about anything.

In Italian, double negatives are perfectly correct and actually required here! When a negative word like niente (nothing/anything) comes after the verb, you must put non (not) before the verb. It literally translates as 'I do not think to nothing,' which means 'I am not thinking about anything.'

Why is the preposition a used instead of a word for 'about'?

The verb pensare (to think) almost always pairs with the preposition a when you want to say you are thinking about someone or something. While a usually means 'to' or 'at', it is helpful to imagine you are directing your thoughts 'to' that thing.

The English translation says 'I am not thinking', but isn't penso just 'I think'?

Yes, penso is the simple present tense. However, Italian uses the simple present for actions happening right now much more often than English does. Because of this, penso can translate to both 'I think' and 'I am thinking' depending on what makes sense in context.

Why isn't the word io (I) included at the beginning of the sentence?

In Italian, the verb ending tells you who is doing the action. The -o ending on penso already means 'I think'. Adding the pronoun io is completely unnecessary unless you want to add special emphasis, like saying 'Well I am not thinking about anything.'