Note that a point (x1, y1) lies on the line l if and only if ax1 +by1 +c = 0.
Suppose that points (α, β) and (γ, δ) are on the same side of l. Then the line segment from (α, β) and (γ, δ) does not cross l. So ax1 + by1 + c≠ 0 for each point (x1, y1) on the line segment. Thus the expression ax1 + by1 + c cannot change sign as the point (x1, y1) moves along the line segment from (α, β) to (γ, δ). Hence the signs of aα + bβ + c and aγ + bδ + c are the same.
Points on different sides of l must have different signs, as it is possible to choose (x1, y1) so that ax1 + by1 + c > 0, and choose (x2, y2) so that ax2 + by2 + c < 0.