Why does knowing the impulse response allow you to determine the output for any LTI system?
(Adapted from this answer on dsp.SE)

The reason that the impulse response (also called the unit pulse response for discrete-time systems) determines the output for arbitrary input $x$ to an LTI system is that


  The output of a linear time-invariant system in response to input $x$ is 
  the sum of scaled and time-delayed versions of the
  impulse response.


This is because of the linear property of the system: if the response to input signal $x_i$ is $y_i$, then the response to input $x_1+x_2+ x_3 + \cdots$ is $y_1+y_2 + y_3 + \cdots$.  So, let's decompose the input signal $x$ into a sum of scaled
unit pulse signals. The system response to the unit pulse signal
$\cdots, ~0, ~0, ~1, ~0, ~0, \cdots$ is
the impulse response or pulse response 
$$h[0], ~h[1], \cdots, ~h[n], \cdots$$
and so by the scaling property the single input value $x[0]$,
or, if you prefer
$$x[0](\cdots, ~0, ~0, ~1, ~0,~ 0, \cdots)
=  \cdots ~0, ~0, ~x[0], ~0, ~0, \cdots$$
creates a response
$$x[0]h[0], ~~x[0]h[1], \cdots, ~~x[0]h[n], \cdots$$

Similarly, the single input value $x[1]$ or creates
$$x[1](\cdots, ~0, ~0, ~0, ~1,~ 0, \cdots)
=  \cdots ~0, ~0, ~0, ~x[1], ~0, \cdots$$
creates a response
$$0, x[1]h[0], ~~x[1]h[1], \cdots, ~~x[1]h[n-1], x[1]h[n] \cdots$$
Notice the delay in the response to $x[1]$.  We can continue further
in this vein, but it is best to switch to a more tabular form
and show the various outputs aligned properly in time. We have
$$\begin{array}{l|l|l|l|l|l|l|l}
\text{time} \to &amp; 0 &amp;1 &amp;2 &amp; \cdots &amp; n &amp; n+1 &amp; \cdots \\
\hline
x[0] &amp; x[0]h[0] &amp;x[0]h[1] &amp;x[0]h[2] &amp; \cdots &amp;x[0]h[n] &amp; x[0]h[n+1] &amp; \cdots\\
\hline
x[1] &amp; 0 &amp; x[1]h[0] &amp;x[1]h[1] &amp; \cdots &amp;x[1]h[n-1] &amp; x[1]h[n] &amp; \cdots\\
\hline
x[2] &amp; 0 &amp; 0 &amp;x[2]h[0] &amp; \cdots &amp;x[2]h[n-2] &amp; x[2]h[n-1] &amp; \cdots\\
\hline
\vdots &amp; \vdots &amp; \vdots &amp; \vdots &amp; \ddots &amp; \\
\hline
x[m] &amp; 0 &amp;0 &amp; 0 &amp; \cdots &amp; x[m]h[n-m] &amp; x[m]h[n-m+1] &amp; \cdots \\
\hline 
\vdots &amp; \vdots &amp; \vdots &amp; \vdots &amp; \ddots  
\end{array}$$
The rows in the above array are precisely the scaled and
delayed versions of the impulse response that add up to
the response $y$ to input signal $x$.

So, we see that if we know the impulse response of an LTI system, we can figure out the response of the LTI system to any arbitrary signal $x$. We don't need any additional information about the LTI system beyond the impulse response.