Transistors are switches, yes, but switches are more than just for turning lights on and off.
Switches are grouped together into logic gates. Logic gates are grouped together into logic blocks. Logic blocks are grouped together into logic functions. Logic functions are grouped together into chips.
For example, a TTL NAND gate typically uses 2 transistors (NAND gates are considered one of the fundamental building blocks of logic, along with NOR):
simulate this circuit – Schematic created using CircuitLab
As the technology transitioned from TTL to CMOS (which is now the de-facto standard) there was basically an instant doubling of transistors. For instance, the NAND gate went from 2 transistors to 4:
simulate this circuit
A latch (such as an SR) can be made using 2 CMOS NAND gates, so 8 transistors. A 32-bit register could therefore be made using 32 flip-flops, so 64 NAND gates, or 256 transistors. An ALU may have multiple registers, plus lots of other gates as well, so the number of transistors grows rapidly.
The more complex the functions the chip performs, the more gates are needed, and thus the more transistors.
Your average CPU these days is considerably more complex than say a Z80 chip from 30 years ago. It not only uses registers that are 8 times the width, but the actual operations it performs (complex 3D transformations, vector processing, etc) are all far far more complex than the older chips can perform. A single instruction in a modern CPU may take a many seconds (or even minutes) of computation in an old 8-bitter, and all that is done, ultimately, by having more transistors.
If you are only confused by the "test particle," then you can think of it similarly to a multimeter. With a multimeter, you can probe a circuit to determine a voltage at one part of a circuit relative to another. With a test particle (or test charge, as I got used to hearing), you place it at a point in space, and "observe" it's behavior to see how electric (or magnetic) fields are oriented.
Like charges repel each other, so if a test charge would tend to move in a certain direction in space, then either that direction contains a negative charge (assuming you use a positive test charge), or the opposing direction contains a positive charge.
The movement of a test charge will always oppose the energy gradient (in three dimensions, energy is a scalar field, so the spatial derivatives are your forces, since energy divided by length is force). Thus, a test charge will move in the direction that achieves it's lowest energy state.
In a vacuum (such as space), ionized particles can move freely. The test charge is usually assumed to be in a state such that it can move freely. This doesn't necessarily correlate to anything real, but is a hypothetical state so the fields can be analyzed easily. You are correct in that circuits don't involve the movement of atoms or positive charges, but rather electrons (due to d-block delocalization, but that's chemistry) move. The positive charges (protons) are held in a crystalline lattice, which is why they don't move. In a conductor, electrons can move freely, so they move in response to an applied electric (or changing magnetic) field.
In free space, however, a test charge (which is really an ion, or a proton, or a positron, or a myriad of other positively charged particles) is not constrained by the bonds that hold metal atoms in place, so it can move in response to an applied field. Specifically, interactions governed by photons cause particles to exchange energy, creating the field gradient mentioned before. Therefore, test charges in free space move similarly to how electrons move in a metal (or another conductor).
While this is a convention (positive charges just seemed to be more reasonable when a lot of this math was derived), you could physically create a positively charged particle and observe it's trajectory. You could even do this at home. You'd be using a Cloud Chamber and a Beta emitter to see them.
I hope this was helpful! Let me know if you need any more clarification.
There are no energy states of transistors in the sense that energy is stored in a transistor, except for the tiny tiny amount of energy stored in the gate charge of a mosfet. Before you latch onto that as a storage device: your average household cat would probably be a more practical energy storage system (in the form of static charge).