A scalar $m$ breaks the gauge symmetry for any gauge field whose left versus right behaviors are different (i.e. chiral fields). If you break $Ψ$ into left and right helicity components $L$ and $R$, then $\bar{Ψ} m Ψ$ is actually $\bar{L} m R + \bar{R} m L$. For dynamics, this entails an oscillation between left and right helicity components.
You can view it as follows. Generically, helicity refers to the component of angular momentum along an axis collinear with the momentum. For bodies moving at under light speed, the direction of motion is relative. To an observer moving the same way as the body is, but at a faster speed, the body appears to be moving the other way and its helicity has the opposite sign; so that helicity is relative in both left/right'edness and magnitude.
For light-speed bodies, there is no overtaking them and no rest frame. So, helicity can be an invariant. In the classification into "tardion", "luxon", "tachyon" (respectively for sub-light, light-speed and super-light systems), luxons has a further sub-classification into generic versus the (unnamed) "helical" subclass. In the latter, angular momentum is entirely on an axis collinear with momentum and helicity is an invariant - both its sign and magnitude. Photons fall in this class. In particular, photons have helicity, not spin.
The only other subclass that has helicity as an invariant are the ones where it is trivially so: "spin zero" bodies, that possess no intrinsic angular momentum and therefore zero helicity.
Here's where the problem occurs: for the weak force, the charge is left-helicity! Up to proportion. For anti-matter, it is right helicity. Charge is an invariant property - which forces helicity to be an invariant property, which forces the bodies to be helical luxons, since that's the only (non-trivial) case where helicity is an invariant.
In the 1950's and before that, they would have summed it up as: "WHOOPS!" and pulled out the red "REJECT!" stamp to plaster on top of the published papers.
The $Ψ$ in the Dirac equation gives the appearance of two light-speed helical states, $L$ and $R$, with the equation giving you an oscillation between the two states, the oscillation taking place at a rate proportional to $m$. In effect, it's like a light speed body that is constantly and randomly zig-zagging between left and right states, whose time-averaged speed and path are that of a tardion with a positive rest mass $m$.
To make this co-habit consistently with a chiral field, like the weak force, you have effectively turn on $m$ as a field by treating it as $m = g φ$, where $g$ is the $φ$-charge of the body and the field $φ$ is permanently and everywhere switched on. So, now the Dirac term $\bar{Ψ} m Ψ$ becomes the "Yukawa term" $\bar{L} g φ R + \bar{R} φ^† g L$. What does $φ$ do? It drives the oscillation between the left and right helicity states $L$ and $R$. The $g$'s can be skewed, so that "mass" eigenstates need not line up with "charge" eigenstates. So, $g$ could actually be a matrix as well: $\bar{L} g φ R$ and $\bar{R} φ^† g^† L$.
So, all the tardions that interact with the chiral field are explained away as actually being helical luxons that only have the appearance of being of positive mass $m$ and slower than light, thanks to the zig-zagging action driven by the $φ$ field. Fundamentally, everything that interacts non-trivially with the weak force is a de facto helical luxon of zero mass. Any appearance of being otherwise is just an illusion brought about by the action of the $φ$ field.
These terms can be made to respect gauge symmetry by endowing $φ$ with suitable transformation properties under a gauge transform. That means the "on-state" for $φ$ can't be unique. Under a gauge transform, it transforms to a different on-state. But only one is selected out as the actual on-state - everywhere in the Universe. Hence, the term "broken symmetry".
For the weak force (which has 3 modes) - and the more comprehensive electro-weak force (which has 4 modes) it is a part of (the other part being the "hypercharge") - one residual mode of gauge symmetry remains; the other three are frozen by the broken symmetry. The forces exhibited by the three frozen modes are short range, because the vacuum is opaque with respect to them, thanks to the interference brought about by the on-state of the $φ$ field.
The fourth mode is long range and the vacuum is transparent with respect to it, since the on-state of the $φ$ also has the gauge symmetry. It also happens to be the one mode that is left-right symmetric. That didn't actually have to be the case. There's nothing fundamental in the model for the electroweak theory that required it. It just is, and that's an unexplained feature.
The long-range force for the fourth mode is the electromagnetic force. It is a combination of the hypercharge and one of the three modes of the weak force. The hypercharge is what actually satisfies the Maxwell equations (and its quantum, the $B$, has zero weak nuclear charge), while the electromagnetic field does not(!) - but rather a non-linear set of field equations that include also the effect of the weak force. Electroweak technically falsifies Maxwell's theory for electromagnetism. The electromagnetic field interacts with the weak nuclear force. In particular, the photon interacts with the weak nuclear force. It's not neutral.