The <PERSON> solution has a simple interpretation. Using your notation, it reads $$\phi(x, t) = \frac{g(x-t) + g(x+t)}{2} + \frac12 \int_{x-t}^{x+t} h(x') dx'.$$ where $g(x)$ is the initial position, $h(x)$ is the initial velocity, and $v = 1$.
Mathematically, the first term above solves the wave equation with initial position $g(x)$ but zero initial velocity, while the second term does the same for zero initial position and initial velocity $h(x)$. By the superposition principle, their sum has initial position $g(x)$ and initial velocity $h(x)$, and is hence equal to $\phi(x, t)$ for all times.
We can understand each of these individual terms by physical intuition.
First consider the term for initial position $g(x)$. We know all solutions of the wave equation are a linear combination of functions of the form $f(x\pm t)$, so the only things we can use are $g(x+t)$ and $g(x-t)$, which both have the right initial position. Finally, we notice that averaging them produces zero initial velocity by the chain rule. 804
Intuitively, if you hold a string and let go, you will make equal-sized waves going in both directions.
Next consider the term for initial velocity $h(x)$. To understand it, consider the following simpler question: suppose $h(x)$ is zero everywhere except for a sharp spike at $x = 0$, corresponding to us hitting the string there at time $t = 0$. What is the resulting shape?
Physically, we expect a 'shockwave' to propagate outward from this event. Solving the wave equation using a similar technique to the one in the previous paragraph, we find $$y(x, t) = \frac12\left( \theta(x+t) - \theta(x-t) \right).$$ That is, everything inside the "light cone" of $(x, t) = (0, 0)$ has been raised up by $1/2$.
For a general $h(t)$, then, the solution is to integrate this $1/2$ over all light cones that could have affected the point $(x, t)$. The limits of this light cone are $x-t$ and $x+t$, yielding the term $$\frac12 \int_{x-t}^{x+t} h(x') dx'$$ in agreement with the formula. 586