How to see your fan speeds on Windows
Turn on Sensor readings in Pulse by Zaweba and your fan speeds appear in RPM in the Fans group on its Temperatures page, if your hardware reports them.

Windows doesn't show fan speeds anywhere in its own settings, but Pulse by Zaweba can. Turn on Sensor readings in Pulse's Settings, or with the button on the Temperatures page, and a Fans group appears on that page listing each fan Pulse can read with its speed in RPM. Whether a fan shows up depends on your hardware reporting it.
Fan speeds sit with the temperatures in Pulse, which makes sense: fans react to heat, so seeing the two side by side tells you whether the cooling is keeping up.
Why it needs a helper
Fan readings come from the motherboard's sensor chip, and Windows only lets programs running as administrator talk to it. Pulse is built to run as a normal user all the time, so instead of asking for full rights it starts a small helper when you ask for sensor readings.
The helper is a second copy of Pulse run as administrator after Windows asks you. It only reads sensors, sends the readings to Pulse over a private pipe every two seconds, accepts no commands and stops when Pulse closes. CPU and motherboard sensors also need the free, signed PawnIO driver, installed separately. Pulse doesn't install the old WinRing0 driver that some monitoring tools rely on.
Turning it on
Open Pulse and go to Settings (Ctrl+5), then switch on Sensor readings. Or open the Temperatures page (Ctrl+2) and use the button there. Approve the Windows prompt. Within a few seconds the Fans group fills in, alongside CPU package and core temperatures and the motherboard sensors.
If no fans appear, your board may not expose them, or the PawnIO driver may not be installed. Laptops in particular often keep their fan control to themselves. That is a limit of the hardware, not something Pulse can get round.
Reading the numbers
Fan speeds vary a great deal between fans, so compare a fan with itself rather than with a figure from elsewhere. Note the speed at a quiet desktop, then again under a heavy load such as a game. A fan that rises under load and falls again afterwards is doing its job. One that reads zero while the PC is warm may have stopped, unless your board is set to let fans rest when cool, which some are.
Look at the temperature tiles at the same time. Each one is coloured normal, warm or hot with its thresholds printed on it. Fans at full speed with temperatures still climbing into hot suggest dust or poor airflow.
Linking noise to a cause
Pulse's History page keeps a list of temperature peaks for today, yesterday or this week, each with the three busiest apps that minute. When a fan was loud earlier, this is where you find out which program made the heat. The temperature alert on the Alerts page can also warn you while it happens, when a sensor stays in its hot band for five seconds.
Pulse by Zaweba keeps everything on your PC and runs light in the tray, so you can leave it watching the fans and temperatures without noticing it's there.