It is a symbol that represents ratios; $A:B$ means "the ratio of $A$ to $B$". (Shows up all the time in Euclid; e.g., Book V.) When we write "$A:B=C:D$", we mean the ratio of $A$ to $B$ is the same as the ratio of $C$ to $D$.
Here he is talking about three ratios; the fact that $x:y:-1 = x_1:x_2:x_3$ means that the vectors determined by $(x,y,-1)$ and by $(x_1,x_2,x_3)$ are parallel. Note that $(x,y,-1) = (x,y,0) - (0,0,1)$ is the vector with starting point at $(0,0,1)$ and end point at $(x,y,0)$; and $(x_1,x_2,x_3-1) = (x_1,x_2,x_3) - (0,0,1)$ is the vector with starting point at $(0,0,1)$ and endpoint at $(x_1,x_2,x_3)$. Since $x:y:-1 = x_1:x_2:x_3$, they are parallel; since they both start at the same point, that means that the line through $(0,0,1)$ and $(x,y,0)$ (determined by the first vector) and the line through $(0,0,1)$ and $(x_1,x_2,x_3)$ (determined by the second vector) are the same, since they are parallel and they both go through $(0,0,1)$.
I don't know about Bergman, but Burris & Sankappanavar: A Course in Universal Algebra has this to say on page 3:
The symbol “$=$” is used to express the fact that both sides name the same object, whereas “$\approx$” is used to build equations which may or may not be true of particular elements. (A careful study of $\approx$ is given in Chapter II.)
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