What can a quantum query to a function do?
the gate $f$ performs the transformation 
$$\rvert x\rangle \rvert0\rangle \to \rvert x\rangle \rvert 0\oplus f(x)\rangle, $$

but $0$ is idempotent for $\oplus$ so you simply get $\rvert x\rangle \rvert f(x)\rangle$.
Now, by linearity, if you input a superposition instead of a basis states, you'll get a superposition of the "answers", e.g.,

$$\frac1{\sqrt2}(\rvert 0\rangle \rvert0\rangle+\rvert1\rangle |0\rangle) \to \frac1{\sqrt2}(\rvert 0\rangle \rvert f(0)\rangle+\rvert 1\rangle \rvert f(1)\rangle)$$

Then, if you start with a Hadamard, and get an equal superposition of all basis states, you end up with a superposition of all possible "answers".
$$\frac1{N} \sum_x \rvert x\rangle \rvert 0 \rangle \to \frac1{N} \sum_x \rvert x\rangle \rvert f(x) \rangle.$$

The fact that $x$ or $0$ are in fact a "string" over $\{0,1\}$ doesn't make any difference - treat them as a single quantum system of higher dimension (or treat each qubit as a separate function $f_i$ where $f_1()f_2()\cdots f_n()=f()$ are the bit-wise functions.