What's the purpose of the fingers. What n=2 did what connect the 2 drains together and also the gates together and also the sources together. So in effect created 2 transistors in parallel? Correct?
Same would have been the case with m=2 right?.
Is this done to increase the current being sourced / sinked?
The purpose for building a transistor out of several transistors in parallel is for matching. If you, for example, want to build a current mirror with a 2:1 ratio of currents, you build the 2x transistor out of two parallel versions of the 1x transistor. This achieves better matching than simply building the 2x transistor 2x wider.
Another reason for this is to control the parasitic, and fold large transistors.
Foling large devices: Say you need a very wide transistor for something like a output driver stage, you might need a transistor that is 30 nm long and 15 um wide. That would be a very, very, very long thin line and just be annoying. So instead, you make it with (for example)100 parallel transistors that are 30 nm long, 150 nm wide.
Parasitics: Say you are making an RF amplifier. In this case you tend to tune out the gate capacitance with an inductor. Your gate poly/metal has resistance, which limits the effectiveness of the inductor and causes losses (and thus less system gain). A longer distance that the current has to travel though the gate results in more resistance. So you can try and make many connections across the length of the gate and drain, but that results in you getting more capacitance (because you are building two plates of a parallel plate capacitor). As a result, you get these kinds of tradeoffs (Source: P. Reynaert, 'Design of High Frequency Integrated Circuits', lecture slides):
EDIT: in case it is not clear what you are looking at: this is a '3D' view of the network to go from the top, ultra thick metal (neon green, top left of each image) to the gate/drain fingers of a transistor (teal-blue thing, bottom right). This is from a process that has, iirc, 9 metal layers + 1 UTM layer.
If you use less fingers, you need this taper to extend for a longer distance (gate-drain capacitance* + gate resistance go up). If you make shorter fingers, it will get too wide (gate-drain capacitance and gate resistance go down, gate-source/gate-substrate goes up).
*Note that the gate-drain capacitance isn't necessarily bad for the speed, but it is bad for stability as it provides a feedback path. In differential mode this can be tuned out (how is a topic for another time) but in common-mode it can't.