Why do valves need high voltage? Is that dangerous?

The way valves work, high voltages are necessary. If they are run on lower voltages, the sound given would become very weak and with lots of distortion.

To make it safe, most modern designs put the high-voltage electronics in a robust metal chassis which is solidly grounded. This way the user is completely protected from the high voltages inside.

Suitable grounding and construction techniques ensure there is no possibility for the case, or any of the input or output terminals, to become live. 

This is important, since some of the bigger amplifiers can have up to 600 volts inside. 

It is advisable that only people specifically trained in high voltage electronics work on valve gear. 

Most modern designs incorporate additional safety features, such as a “bleed-down” facility, where once it’s powered off, the remaining high-voltage charge inside drains away over about 60 seconds.

Some legacy valve gear had exposed high voltage parts, so it pays to be very careful around older valve gear.