What Is the Significance of a Large Residual When Applying Principal Component Analysis?
PCA dimensionality reduction seeks for a linear transformation, mapping the data to a lower dimensional space. The linearity comes from the mapping, which is simply a rotation (and translation if you take into account the de-meaning). In the end, it is nothing but a projection of the data.

So if your low dimensional embedding is not reconstructing the data truthfully, then this means that your data is not suitable for such linear mapping, i.e. not linearly separable. You should look into other techniques such as linear discriminant analysis (canonical variates), Kernel-PCA, which applies kernel trick to achieve non-linearity or manifold learning.