After reading what I could find about the CLIC accelerator, I still don't understand how it would work. If someone could explain how the CLIC would work, I would greatly appreciate it.
Although you have already found the Conceptual Design Report (CDR), I think that a brief summary of the fundamental concepts can be appropriate.
The Compact Linear Collider (CLIC) is a unique design for a $e^+e^-$ collider up to 3 TeV. The only competitive project is the International Linear Collider (ILC) which however aims at a lower energy (1 TeV). CLIC has a number of components, but I assume that your doubts are related to the main linacs.
The choice of normal-conductive accelerating structures (as opposed to the superconducting structures of ILC) has been made for them, motivated by the higher accelerating gradients that can be achieved, therefore making the machine "compact" or allowing collisions at higher energies. When pushing the gradient to very high values one encounters two limitations: breakdowns and ohmic losses. Both of the issues are assessed by shortening the pulse length which means filling the cavities with the Radio Frequency (RF) for times no longer than some hundreds of nano seconds (of course the beam must be matched to this).
The production of such a short and intense RF pulse is not feasible with conventional RF generators (klystrons) keeping acceptable power losses and component number. CLIC assesses this issue in the Drive Beam, where a longer beam, more suitable for conventional acceleration, is produced and repeatedly "folded" on itself to compress its time duration while increasing the intensity. This beam is then decelerated extracting its power in the RF form, suitable for the acceleration of the colliding beams.
The Drive Beam and the Main Beam runs in parallel structures called Two Beam Modules (TBM) equipped with a number of RF connections but no active high-power devices, which simplifies the installation in the tunnel (there is no need for a klystron gallery).
Here is a picture (that you won't find in the CDR) of the two TBMs currently installed at CTF3 (CERN). The module of CLIC is a bit different, but does the same job.
Summarising, the power flow in CLIC is:
Wall -> RF (long pulse) -> Drive Beam -> RF (short pulse) -> Main Beam