First of all, the scheme of the CERN accelerator complex you posted contains not only that single chain which brings the protons to the LHC, but also several other chains which are used for many lower energy experiments conducted in parallel at CERN.
But let's focus on the LHC accelerator chain: why do we need several successive accelerators instead of a single one? The answer is very intuitive from the engineering point of view: technologically, it is much easier to construct several devices specialized for different but limited ranges of some physical parameter than to build a single device that exhibits excellent performance in the entire vast range.
An example in transportation: cars are good to travel at lower speed, airplanes are good for traveling fast. It is exceedingly difficult to build a vehicle that would have equally good performance in the entire range of speeds from 1 km/h to 1000 km/h. Another example is given by professional acoustic systems which usually include several speakers optimized for different frequency ranges.
The same thing for the accelerators. Here the key point is not the energy of the protons per se, but the quality of the beam at a given energy. Accelerating protons is (technically) a minor issue; the major issue is to keep the beam safe and well-behaved. If you dig a bit into the CERN website, you'll soon realize that these accelerators have quite distinct features and each of them was optimized for its own energy range and its own purposes. Some of them serve to accumulate particles, other are specialized in breaking the beams into bunches. At each stage you need to cool the beams down, dump oscillations, etc., and this is done in a different way at different energies.
In principle, you could think of an accelerator that would occupy the LHC ring and would take the protons at a very low energy (say, 1 MeV) and accelerate them up to multi-TeV. Since the protons must always circulate in the same ring, the magnetic field must be adjustable (with very high precision and very high spacial homogeneity) from few microtesla to several tesla, in the range of six orders of magnitude. The beam monitoring system must also be adapted to accurately measure the proton currents differing by 6 orders of magnitude. The similar requirements are placed on the beam steering, orbit correcting and focusing magnets, on the kicker magnets, and on many other components of the accelerator. In short, although technically possible, it would be just too difficult and too expensive. A sequence of several accelerators (which existed before the LHC anyway) is a preferred solution.