Why does a sign difference between space and time lead to time that only flows forward?
We can move back and forth in space, so why does the negative sign mean we can't move back and forth in time?
As illustrated in the answer by Ben Crowell and acknowledged in other answers, that relative sign doesn't by itself determine which is future and which is past. But as the answer by Dale explains, it does mean that we can't "move back and forth in time," assuming that the spacetime is globally hyperbolic (which excludes examples like the one in Ben Crowell's answer). A spacetime is called globally hyperbolic if it has a spacelike hypersurface through which every timelike curve passes exactly once (a Cauchy surface) . This ensures that we can choose which half of every light-cone is "future" and which is "past," in a way that is consistent and smooth throughout the spacetime.
For an explicit proof that "turning around in time" is impossible, in the special case of ordinary flat spacetime, see the appendix of this post: https://physics.stackexchange.com/a/442841.
 Pages 39, 44, and 48 in Penrose (1972), "Techniques of Differential Topology in Relativity," Society for Industrial and Applied Mathematics, http://www.kfki.hu/~iracz/sgimp/cikkek/cenzor/Penrose_todtir.pdf
 Page 4 in Sanchez (2005), "Causal hierarchy of spacetimes, temporal functions and smoothness of Geroch's splitting. A revision," http://arxiv.org/abs/gr-qc/0411143v2
The sign that appears in the metric or line element, i.e. in
$ds^2 = -dt^2 + dx^2 + dy^2 + dz^2$
does establish a difference between space and time, but it does not, on its own, contain all of the physics related to time. For one thing, it does not determine which direction is future and which is past. That direction is established by other considerations such as entropy increase. The other main ingredient here is the claim that worldlines are timelike not spacelike. This really amounts to a statement about conservation laws. We identify a sequence of events along a certain line in spacetime as a sequence associated with one particular entity, such as a particle or a body, because there is something in common at the events: a certain amount of electric charge, for example, or energy and momentum. So we say we have a particle (or larger body) and that is its worldline. A sequence of events along a spacelike line, on the other hand, often doesn't show that kind of common property, so we don't find it helpful to suggest that the same entity was present at all the events. As you see, we are getting quite close to metaphysics here.
how does a relative sign difference lead to a situation where time only flows forward and never backward? We can move back and forth in space, so why does the negative sign mean we can't move back and forth in time?
It is not only the sign difference, but also the fact that there is only one dimension of time while there are multiple dimensions of space. Because there is only a single dimension of time a surface of constant proper time forms a hyperboloid of two sheets. One sheet is future times and the other sheet is past times, so there is no way to smoothly transform a future time into a past time. Future and past are geometrically distinct.
In contrast, because there are three spacelike axes a surface of constant proper distance forms a hyperboloid of one sheet. So you can smoothly transform up into down and so forth. Different spacelike directions are not geometrically distinct.