Basically, linear projections are mapping a high-dimensional space that is flat on all axes to a lower-dimensional space that is flat on all axes. So, if you wanted to represent stuff stacked in a cardboard box on a line, you could project the box's space to a line. Or you could map the Mona Lisa to a line. It's useful in machine learning when one wants to visualize data or simplify a high-dimensional problem, and factor analysis/PCA are based on this (PCA finding naïve coordinates/mappings and factor analysis considering more a priori assumptions on the mapping and spaces).