The motivation for covert movement is that we see some languages that have certain types of movement--like, for instance, movement of wh phrases to the left edge of a sentence--but that other languages do not have that kind of movement. Regardless of the language, we see a motivation for that movement to happen in order to derive certain facts about interpretation or meaning of a sentence, such as wh-phrases moving to signal that a CP is a wh-question, or to create certain scope properties when there are multiple quantifiers in a sentence.

When we need movement to happen in order to model certain interpretive properties of a sentence, but we do not see that movement in the actual language forms, we propose that the movement is "covert". In our model of grammar, this means that that movement happens only after the point of "SPELLOUT", where our derivation splits between the representation that ends up at the level of Phonological Form (which determines the form that is actually pronounced or recorded), and the representation that ends of at the level of Logical Form (which determines what that pronounced form actually means, ie, how it is interpreted).

It's worth noting that this means that if a sentence in English and a sentence in Mandarin Chinese have the same meaning, then the forms of those sentences that end up at LF will have precisely the same structural representation.