I was wondering how neutrino oscillations are possible and lepton number is conserved, as for example in the decay of an anti-muon: $$\mu^+ = e^+ \nu_e \bar{\nu_{\mu}}$$ all lepton numbers are conserved, but let’s now imagine the anti-muon neutrino oscillate to an anti-tau neutriono, then we finally have: $$\mu^+ = e^+ \nu_e \bar{\nu_{\tau}}$$ in which lepton number isn’t conserved, what is the theoretical explanation for this? Or do people just say that oscillations violate this conservation law?
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It is indeed a violation of two of the three separate lepton number conservation laws. You've encountered an example of an approximate conservation law. The cause of these vary.
Terms in the Lagrangian either do or don't provide an opportunity for such transitions. The full Lagrangian $L_\text{total}$ can be thought of as a slight expansion around a first-order approximation $L_0$ which, were it exact, would prevent neutrino oscillation. Let's write $L_\text{total}=L_0+g\lambda$, with $\lambda$ a term that can create such interactions, so for small $g\ne0$ violations are possible but rare. (This is something of a simplification, as "rare" events tend to be more common at a suitable energy scale. Approximate conservation laws are often "valid" in a domain avoiding such a scale.)
It's even possible, though this is still under study, that some interactions may not even conserve the sum $L$ of all three lepton numbers (although your example does). It's possible the exact law only conserves $B-L$ ($B$ the baryon number) instead of $B,\,L$ separately.
