The purpose of PCA is to solve inverse problems and yes it is related to SVD. Let me give you a simpler idea to work with. Imagine you were a policy maker and believed that salaries were correlated with knowledge, so you get a data set of wages and academic attainment. The problem with this set is that it doesn't give you a measure of knowledge, but rather how long someone continued in school. Academic attainment should be correlated with knowledge, but it cannot rationally be a measure of knowledge. Some people who are brilliant end up with low levels of achievement and some people who are not so bright go beyond what would be expected to be achieved for the level of knowledge.
Knowledge is a latent variable to the measured variable.
Now let us imagine we have multiple observed measures for many different things and these things are correlated. What we ideally want to do is find the underlying true features that drive the model, the latent variables, and not the incidental data we can observe.
Principal components analysis allows us to do two things. First, it serves as an estimate of the driving hidden features. Second, it gives a weighting of the importance of those features so we can remove those features that are probably incidentally observed and not really important to the model at hand.