How do power stations maintain 50 hertz?
From UK's National Grid:

Figure 1. The graph shows Frequency data to a 15 second resolution over the hour up to 2016-05-21 13:31 BST.

System frequency is a continuously changing variable that is determined and controlled by the second-by-second (real time) balance between system demand and total generation. If demand is greater than generation, the frequency falls while if generation is greater than demand, the frequency rises.
National Grid has a licence obligation to control frequency within the limits specified in the 'Electricity Supply Regulations', i.e. ±1% of nominal system frequency (50.00Hz) save in abnormal or exceptional circumstances. National Grid must therefore ensure that sufficient generation and / or demand is held in automatic readiness to manage all credible circumstances that might result in frequency variations.
There are two types of Frequency Response Dynamic and Non Dynamic Response. Dynamic Frequency Response is a continuously provided service used to manage the normal second by second changes on the system. While Non Dynamic Frequency Response is usually a discrete service triggered at a defined frequency deviation.

To answer your questions:

So I understand that the generators at the power stations need to spin at exactly 3000rpm.

Not constantly. Over a day it averages to exactly 50 Hz. This used to be a requirement to keep all mains powered clocks in synch. I have one on my electrical supply day/night meter.

This is true no matter what source is (nuclear, coal, gas, hydroelectric etc).

Yes, they all run in synch.

Let's use steam as an example here. How do they use the steam to spin the generator at a constant 50 hertz? It doesn't seem plausible to control the speed with the amount to steam passed though to that precise speed. So how do they do it?

It is, in fact, possible, provided there is enough steam being generated to supply the peak load. With a single steam generator supplying an island, for example, the scheduler would plan ahead using weather information, TV schedules (the famous 10 million kettles going on in the big-game half-time), large load user schedules, etc., to have the thermal plant generating enough steam in anticipation. Meanwhile, while demand is low, the plant has to vent the excess steam to atmosphere. Yes, this is wasteful.
In practice, most grids have a mix of base-load thermal plant with other fast response generators such as gas turbine and hydro which can be switched in and out quickly.
The UK also has capability to import power from France and Ireland by underwater DC interconnects. These allow connection of multiple national grids without synchronising problems.