The graph of a linear function is a straight line

Compute the rise and run between the two points on the graph: rise $=y_2-y_1=(ax_2+b)-(ax_1+b)=a(x_2-x_1)$, using the fact that both points satisfy the same rule $y=ax+b$.

Since $x_1\ne x_2$, we may divide both sides by the run $x_2-x_1\ne 0$: $$\dfrac{y_2-y_1}{x_2-x_1}=\dfrac{a(x_2-x_1)}{x_2-x_1}=a.$$

This computation used only the values $x_1,x_2$ and the rule $y=ax+b$, and the $b$ term canceled out completely — so the ratio equals $a$ regardless of which two points $(x_1,y_1),(x_2,y_2)$ were chosen. A curve on which the slope between every pair of points is the same fixed number is, by the geometric definition of straightness, a line; hence the graph of $y=ax+b$ is a straight line of slope $a$.

Conversely, given any non-vertical line with slope $a$ passing through a point $(x_0,y_0)$, setting $b=y_0-ax_0$ produces a linear function $y=ax+b$ whose graph is exactly that line, which establishes the converse direction of the correspondence between linear functions and non-vertical lines.