The effect of the increase in the number of transistors in a chip on a processor's peformance
Transistor counts, in the end, is the limiting factor of functionality. You can have more functionality when you have larger transistor count than when you have not.

Primarily, increased transistor count is always good for performance. Many algorithms (such as multiplication) benefit from transistor count immensely - when you've got "unlimited" transistors you can multiply large words in a 1 to 3 cycles as opposed to cycle counts on the order of bit numbers in words (to give one example). You can also uncover instruction level parallelism by doing multiple non-dependent instructions simultaneously (superscalar execution), bundle more cores (coarse parallelism) and vector computation accelerators (such as SSE); being able to store working data close to CPU is also beneficial (thus, caches).

The problems with many transistors are also obvious. The 2 most important ones are:


Power consumption
Signal (especially clock) distribution. The more transistors there are, the higher are the chances that something will try to get out of sync.


For a good clear cut example on "transistor count" benchmarking try to look at cycle count of particular instructions and other "transistor expensive" CPU metrics:


How many cycles it takes for a CPU to complete complex instructions. I already mentioned multiplication (which is critical for game performance). Floating point instruction cycle counts are also indicative (single vs double precision, etc.). The basic rule is - the faster arithmetic instruction executes, the more transistors were put to implementing it.
How many execution units a given CPU has (how many adders, multipliers, load-store units)? Each of this costs transistors. Also, check how many instructions a given CPU can decode/dispatch/complete per cycle (all those features are very transistor costly).


You don't even need to go past this - just compile a table of the above metrics for CPUs from various generation and superimpose those over some long living benchmark figures. Try to compute "performance per CPU cycle" value to negate the effect of rising clock frequencies.