Maybe this plot of the gravitational field will help:
EDIT re the comment:
First of all you mean electric field lines, not magnetic. And yes, they do look the same because they are the same. The underlying field equations are identical (in the static nonrelativistic limit). The difference is that for gravity like charges attract whereas for electric forces like charges repel.
Maybe the plot is a little less than clear. Keep in mind this plot is the force on the fluid, not the bubbles. You were meant to take away that the fluid in between the bubbles flows out of the space and the bubbles get closer. ;)
To really do this problem properly you need some assumptions about the fluid: namely that it has surface tension to stabilise the bubbles and also that the flow is incompressible (otherwise you need to keep track of the density everywhere and it gets truly awful). You also need to put the system in a box (it can be a gigantic box - the size of it doesn't really matter in the end) just to avoid the ambiguities of having an infinite mass of fluid. Under these assumptions you can argue that the bubbles can't expand, and also any flow out of one region has to be balanced by a flow into another region.
I'm guessing you're not at the point of seeing field theory yet. Take this as an illustration that there are complicated ways of doing simple problems. The advantage of field theory is that it is much more general and powerful for other problems. But for this you don't really need it - the "negative mass" argument gives you the right answer. But this may give you confidence that the argument given about "negative mass" is correct. In fact, this is better because we don't need to invoke "negative mass" at all - we just talk about the fluid.
If you assume, based on the above, that the bubbles don't change shape or size, then you can treat the problem very simply. All you need to know is that the gravitational potential of a bubble is
$$ \phi(r) \propto \frac{1}{r}, $$
with a positive sign outside the bubble and
$$ \phi(r) = \text{a constant}, $$
inside the bubble. If you know Poisson's equation for the gravitational field you can derive this. You get the total potential by adding the potentials created by the two bubbles. You also need to know the gravitational energy a little parcel of fluid of volume $\Delta V$ is given by
$$ \Delta U = \phi(x,y,z) \rho(x,y,z) \Delta V, $$
where $\rho(x,y,z)$ is the density, which is constant everywhere outsite a bubble and zero anywhere inside. You add up the energy for every piece of fluid ("do an integral") to get the total energy, and see if it increases or decreases with respect to increasing the seperation between the bubbles. The system will naturally go in the direction that decreases the total energy.
So what does the energy look like? Here (energy units are arbitrary, length units are in bubble radii):
At seperations less than 2 radii the bubbles are intersecting so you can no longer trust the calculation - surely the bubbles start changing shape when they collide! But for larger seperations the calculation is ok and look, the energy increases. So it costs energy to pull the bubbles apart. The bubbles are attracted to each other!
The fact that nearly the same calculation done for electromagnetism gives charges which repel each other is an interesting and important clue to the difference between the two theories. If you are really super keen you might have a go at that problem one day.