What does it mean that forces are carried by virtual particles, since virtual particles are just math? According to PBS Spacetime and other reputable sources, virtual particles are really just a handy math trick that make it easier to solve certain problems in quantum field theory, but aren't strictly necessary even for this and shouldn't be thought of as physical objects.  OK, but then why do we talk about virtual particles being the carriers of forces?  What's actually happening in the relevant fields that causes forces between particles, since the virtual particles don't directly correspond to something physical?  For example, when people talk about the electromagnetic force being "carried" by virtual photons, what does this actually mean physically?  Obviously forces exist, as we can measure them, so something which is purely a mathematical convenience can't be responsible for them.
 A: There are several ways to think about virtual particles and what they "physically mean". The simplest and probably most satisfying explanation comes from the energy-time Heisenberg uncertainty relation: $$\Delta E \propto \frac{1}{\Delta t}. $$
Essentially, the virtual particles are so short-lived that their energies cannot be well specified.
A more detailed explanation comes from the perturbative calculations that underly Feynman diagrams. In a QED Feynman diagram, a virtual photon is a graphical representation of the photon propagator: $$ \frac{-i g_{\mu \nu}}{p^2 + i \epsilon}, $$ which is a factor in the expression for the matrix element represented by the diagram. In the propagator, all values of momentum of the virtual photon, $p$, are allowed. Thus, even though momentum conservation is obeyed by the diagram, the relativistic mass energy relation, $E^2 = p^2 + m^2$, is not enforced for the virtual photon--it's off-shell.
This is not a concern because the virtual particle never travels far and is not a "free particle" which, by definition obeys the relativistic mass-energy relation.
The reason we must include virtual photons is because the structure of the QED Lagrangian forbids certain types of vertices in Feynman diagrams, and thus limits the ways that we can draw a given scattering process. For example a 4-electron vertex is forbidden by the QED Lagrangian. Therefore, since the only allowed QED interaction vertex is 2 electrons and one photon, we must draw electron-electron scattering with a virtual photon in the middle:
To answer your question about virtual particles "carrying force", it's important to understand that the conventional "force is action at a distance" is fundamentally not true in quantum field theory.  For something to happen, particles must interact locally--represented as a vertex of a Feynman diagram.
Due to the structure of the QED Lagrangian and the allowed diagrams, we can state that the photon is the gauge boson which mediates or "carries" the electromagnetic force because it is exchanged by electrically charged particles. This force can only be present if a photon is exchanged--and the only type of photons that can be exchanged in this manner in a scattering process are virtual.
A: 
Obviously forces exist, as we can measure them, so something which is purely a mathematical convenience can't be responsible for them.

The concept of "force" at the level of particle interactions comes in the form of $F=dp/dt$ where $p$ is the momentum involved in the interaction. The virtual particle concept comes tied up with the iconic representation of Feynman diagrams in quantum field theory  which guide the way quantum numbers and conservation laws are retained in the calculations of an interaction.
Take this simple interaction:

an electron scattering off an electron. The crossection can be calculated following the Feynman rules for turning diagrams in calculable integrals.
The momentum change in the scattering is modeled by the internal line which is labeled as a virtual mathematical line. This line has also all the quantum numbers associated with the photon except the mass which should be zero for a real four momentum attributed to a photon. That is why the internal lines are called virtual. They carry the quantum numbers of a particle but the four vector in the calculation is off mass shell.
To answer the title :

What does it mean that forces are carried by virtual particles, since virtual particles are just math?

It is a sloppy way of saying the above explanation  in a few words.
