Why is it said that maximum likelihood becomes intractable if there are latent variables? It is said that EM algorithm helps in cases where direct MLE cannot be carried out due to missing/latent variables. However, I could not understand why direct MLE cannot be carried out when there are latent variables.

Is it something related to the term f(xi|theta)?
My Understanding:
Data point xi is actually a collection of variable values. Example: if data point xi is house-metric, variables length, breadth, and height will define it: xi=(400,500,200). In case there is a latent variable V that affects xi, V cannot be properly defined in xi since V cannot be observed. So, due to that xi will not be complete and f(xi|theta) cannot be determined. Hence, MLE becomes intractable.
Is my understanding correct? Or is there a different / additional reason.

Your understanding is correct that we cannot observe the latent variable, therefore we need to use the EM algorithm. In the E step we estimate our distribution for the latent variable and In the M step we estimate the parameters that will minimize the KL divergence.
To be more clear you can think of f(x|0) in your question, which I believe is the likelihood, as a joint distribution of f(x,V,0). So, your term would be like f(x|V,\theta)f(V|\theta). So, x is now dependent on V which you do not observe.
Hope this helps.
