I understand that scalar invariance, in the context of Structural Equations Modeling (SEM), is having the intercepts for observed variables loading on the same latent variable be invariant across multiple groups. However - what does scalar invariance mean substantially? What are its implications?
It means that for the same score on the latent variable, people in the two groups do not have different intercepts on the observed variables.
Say you're comparing two racial/ethnic groups on a measure of ability that's used in job selection. You find that you don't have scalar invariance for one item. That means that one group finds one question easier than the other group. That means that if you take the total score, you're going to get a biased score. There's an example of that here: https://www.talentqgroup.com/media/84831/policy_assessment_and_the_law-march-2013-.pdf (look at the British Rail example).
Second example: You want to measure depression, so you ask about crying. Women cry more than men, whether they're depressed or not. Women are therefore going to get higher scores on the measure of depression, even if they're equally depressed.