They are mathematical constructs, but they can sometimes lead to viable intuition. Classic (and necessarily simplifying) "how it works" explanations for Hawking radiation, the Casimir effect, deep inelastic scattering results, and more rely upon invoking virtual particles.

Matt Strassler was among several theorists who helped me, as a graduate student, to develop a deeper intuition for quantum fields -- it is important to develop an understanding of the strengths and weaknesses of both the simplistic and nuanced ways of thinking about and discussing these interactions.

It is tempting to regard Feynman diagrams as if they are actually what is happening. One should only use them as guides to intuition. They are actually compact expressions of successive terms in a perturbative expansion, terms that can interfere with one another sometimes. Theorists (like Matt), who work with these things every day come to develop a more-nuanced understanding, generally one that places greater weight on fields (not just the bosonic force-mediating fields, but the fields for the fermions, too) than understood by even many practicing physicists in other specialties.

Another helpful insight that may aid understanding of virtual particles as you start to understand things more-deeply: the photons we observe with our eyes from distant stars need not be exactly on-shell, just very (very!) close.