Why do Italians respond to 'grazie' with 'di niente'?
Because it means
You have nothing to thank me about
which is somehow equivalent to the English expressions
Don't mention it
Not at all
and so on.
The idea is minimizing the importance of what the person is thanking you for, letting her understand it wasn't a hassle for you, therefore resulting in a polite expression.
Other alternative forms
Non c'è di che
the latter literally expressing the concept Non c'è di che ringraziarmi
Spanish and Portuguese, as well as French, do the same
As suggested by I.M. in the comments this is also true in German