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I am hoping someone can clarify how parameter estimates for structural equation models (SEM) are usually interpreted in practice.

By that I mean, suppose we have an SEM of the form

$$ y = \mu + \Lambda \omega + \epsilon $$

$$ \eta = \Gamma \xi + \delta$$

where $\omega = [\eta, \xi]^T$ are latent variables, and $\epsilon, \delta$ are random noise terms.

So we obtain estimates for the coefficients $\lambda_{ij}$ and $\gamma_{ij}$ that comprise the matrices $\Lambda$, $\Gamma$ respectively, and want to make inferences on the relationships between the observed variables $y_i$, the latent explanatory variables $\xi_i$, and the outcome latent variables $\eta_i$.

So the structural equation is straightforward and can be interpretted as with any regression model. Eg. $\gamma_{ij} > 0$ means $\xi_j$ has a positive effect on $\eta_i$.

Where I'm confused is the measurement equation. If we interpret it as we normally do with regression models, we would describe the effect the unobserved latent variables $\omega$ have on the observed variables $y$, which doesn't really make sense to me, since $\omega$ is unobserved?

Since the measurement equation is intended to group observed variables into unobserved latent variables, would it make more sense to instead talk about correlation?

For example, if $\lambda_{ij} > 0$ we might say that $y_i$ and $\omega_j$ are positively correlated. Does that make sense?

Basically, I just want to know how to interpret the measurement equation in a way that makes sense given the explanatory variables are not observed. Hopefully I've explained my confusion clearly.

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See Borsboom et al (2013) [DOI: 10.1037/0033-295X.110.2.203] for an excellent discussion on the meaning of latent variables and components of measurement models. This should be required reading for anyone considering latent variable models.

Broadly, structural equation models are causal models, and causal assumptions are laden in the path coefficients and the absence of estimated paths. A truly agnostic interpretation of factor loadings is that a one unit change in the level of the latent variable is associated with a $\lambda_j$ change in the expected value of the indicator. Typically, though, if we want to interpret latent variable models as measurement models, we need to invoke a causal interpretation of the factor loading and claim that a change in the latent variable causes a change in the expected value of the indicator. The validity of the causal claim is subject to critique, but that is what the process of validating measurement models and instruments is designed to protect against.

A more philosophical point is whether latent variables exist and whether it makes sense to talk about incrementing the value of a latent variable by one unit. By entering it into a model and estimating its mean and variance, you are implicitly requiring the latent variable to be a continuous variable that has the potential to vary. Even if the measurement model hasn't proven to you that the latent variable "exists", the fit of your model implies that a model that involves a continuously varying latent variable that has the same causal functions as an observed variable fits better than a model lacking such a feature. The interpretation of the reality of the latent variable may follow from the acceptance of the statistical interpretation of the model. You couldn't substantively interpret any path in the model without interpreting the paths emanating from the latent variable as you would paths from an observed variable.

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