Understanding $\lvert z-1 \rvert+\lvert z+1 \rvert=7$ graphically $\lvert z-1 \rvert+\lvert z+1 \rvert=7$ is a circle of radius 3.5 if we use a computer algebra system to draw it. One can take $z:=x+iy$ and get the equation
$$\sqrt{(x-1)^2+y^2}+\sqrt{(x+1)^2+y^2}=7$$
Then we can square both sides and get another expression, but from that expression we still won't likely read the said circle.
So how can we actually understand, without using a computer algebra system, that $\lvert z-1 \rvert+\lvert z+1 \rvert=7$ represents the said circle? I'd guess this has something to do with the average equidistance of $z$ from the points $1$ and $-1$.

Although your example is mistaken, I still think the general question is useful.
One observation is that $|z-z_0|$, for any constant complex value $z_0$, represents the distance of $z$ from $z_0$.  For instance $|z-1|$ represents the distance of a complex number (in the complex plane) from $1$.  So the equation
$$
|z-1| = 3
$$
is the equation of a circle (again, in the complex plane) of radius $3$ and centered on the complex value $1$.  Once one internalizes this sort of thing (and knows the definition of an ellipse as the locus of points with the distance sum property), the equation
$$
|z-1|+|z+1| = 7
$$
can be read off quite straightforwardly as an ellipse with foci at $-1$ and $1$.
