The Wikipedia entry for scale invariance states
In statistical mechanics, scale invariance is a feature of phase transitions. The key observation is that near a phase transition or critical point, fluctuations occur at all length scales, and thus one should look for an explicitly scale-invariant theory to describe the phenomena.
I assume the implicit rationale behind "near a phase transition or critical point, fluctuations occur at all length scales" is that the at the critical point, a system's correlation length diverges.
In general, the correlation length will not be the only scale within a theory, though. There could be masses, charges, etc. that introduce a scale as well.
Why is it still a universally valid statement that physics becomes scale independent at phase transitions?