The Osher paper does define what a weak solution is. We seek a solution $w$ of $x$ and $t$ such that
$$ \partial_t w + \partial_x f(w) = 0 $$
for a known function $f$ (the flux function), given initial conditions
$$ w(x,0) = w_0(x) $$
for known $w_0$, for $-\infty < x < \infty$ and $0 < t < \infty$. A weak solution is
a bounded measurable function $w$, such that for all $\varphi \in C^\infty_0(\mathbb{R} \times \mathbb{R}^+)$
$$ \iint\limits_{\mathbb{R}\times\mathbb{R}^+} (w \varphi_t + f(w) \varphi_x) \, \tag{1.2 a}\mathrm{d}x\,\mathrm{d}t = 0, $$
$$ \lim_{t\to0} \lVert w(x,t)-w_0(x) \rVert_{L^1} = 0. \tag{1.2 b} $$
That is, we want to admit a broader class of solutions than just those that we can plug into the original equation. In particular, the original equation indicates $w$ should be differentiable, given that we are differentiating it. However, this would not allow us to modal shocks, which are simply discontinuities in the solution. Shocks are physically allowed in continuum fluid mechanics, and so we seek a mathematical framework in which they make sense. At the same time, we want to preserve boundedness (the quantities don't go to infinity on the domain) and measurability (the solution should be describable in terms of measurable sets, so it's not doing something weird).
The framework we choose is that of the weak derivative. If I denote weak derivatives with bars, then what we do is replace the original equation with
$$ \bar\partial_t w + \bar\partial_x f(w) = 0 $$
Integrating against an arbitrary smooth (in $x$, continuous in $t$) test function $\varphi$, we know solutions to this equation must satisfy
$$ \iint\limits_{\mathbb{R}\times\mathbb{R}^+} \big(\bar\partial_t w + \bar\partial_x f(w)\big) \varphi \, \mathrm{d}x\,\mathrm{d}t = 0. $$
By definition of the weak derivative, this is equivalent to (1.2 a).
