the solution surface
It helps to picture a solution of a first-order PDE not as a formula but as a shape. If u = u(x,y) solves the equation, then the points (x, y, u(x,y)) trace out a surface floating above the (x,y)-plane — a landscape whose height at each spot is the solution value. This is the solution surface, and the whole method of characteristics is really a way of building this surface curve by curve.
Here is the geometric heart of it. For a u_x + b u_y = c, rearrange to a u_x + b u_y - c = 0, which says the vector (a, b, c) is perpendicular to the surface's normal (u_x, u_y, -1) at every point — equivalently, (a, b, c) lies tangent to the surface. So the characteristic curves, whose tangent is exactly (a, b, c) from the characteristic ODEs, lie entirely within the solution surface: they are threads embroidered onto it. To construct the surface you take the initial data curve (a thread already drawn in space), and through each of its points run a characteristic; the family of characteristics sweeps out the whole surface. The solution surface is woven from characteristics.
This picture makes several truths obvious. Why data on a characteristic fails: it gives you only one thread, not enough to sweep a surface. Why crossing characteristics break the solution: the surface would have to take two different heights over one point in the plane, which no graph u(x,y) can do — the surface folds over and the single-valued solution dies. Thinking in terms of the solution surface turns analysis into geometry.
For u_x + u_y = 0 with u(x,0) = g(x), each characteristic is a line x - y = constant carrying the constant value g of that constant. Stacking these horizontal threads at their proper heights builds the tilted surface u = g(x - y) hovering over the plane.
Lay the characteristics at their right heights and the solution surface appears, thread by thread.
A solution surface is single-valued by definition — exactly one height above each (x,y). When characteristics cross, the surface built from them folds and stops being a graph; that is the geometric face of solution breakdown, and the cue to switch to weak solutions.