Let’s assume you have a modern SLR with a beam-splitting mirror in the camera body. The purpose of a beam-splitter in a camera is 3-fold. First, a portion of the incoming light will be split to the viewfinder so you can view the scene. Second, a small portion is split to the AF system so that the camera can determine focus. Third, yet another portion of the incoming light is split to the meter system for determining exposure. This split (viewfinder, AF system, meter system) has to be constant, say 60%, 20%, 20% (although we don't really know the exact values).
A linear PL is comprised of a single layer of polarizing film between 2 layers of glass. With a simple PL, phase & angle changes cause the split ratio to vary. Sometimes 20% is split to the metering system & another time the metering system may receive 15%. In this case the metering system will be inaccurate, because it doesn’t know the % of light received. It’s possible for the discrepancy to get rather large.
The AF system faces a similar problem. Most AF system require f/5.6 or faster to operate properly. If 20% of the coming light is just enough for the AF system to work, & the AF system sometimes only receives 10% of the incoming light, then the AF system may fail because there isn’t enough light for it to operate properly.
A CPL solves this problem with an additional layer of polarizing film known as a quarter-wave plate. It transforms linearly polarized light passing though the 1st layer into circularly polarized light. Circular polarization eliminates phase & angle variations, & interacts with a beam-splitter in the same way as unpolarized light.