It means that the MCAR hypothesis is rejected, i.e., the missing data are not missing completely at random and most likely the probability of missingness depends on other variables in the model (i.e. it is not constant across all observations).
More information on that particular test is available in
Fuchs (1982) Maximum Likelihood Estimation and Model Selection in Contingency Tables With Missing Data J of Amer Stat Assn.
Also, I forgot to say, you don't need to worry about this. The estimates of the LCA model are still unbiased even though MCAR doesn't hold. That is because the ML estimation yields unbiased estimates under the more general MAR hypothesis.