How were random numbers generated before computers existed?

Before computers existed, random numbers were generated by using a "random number table". One of the formats of random number table is shown in the figure below.

before computers, why would you want one ????

It is true that you can get tables of random numbers but that just begs the question of where did the tables come from.There are ways to use thermal noise measurements to generate random numbers but pre electronics I expect it came down to specialist dice. You can make pretty much any number of equally likely faces on a die by making it like two cones glued together at the bases, then carve each cone into identical triangular faces. When there isn't an obvious top face, you use the bottom face instead.
  
But as it happens, an icosahedron (20 sides) is one of the platonic solids, so it can generate base ten random numbers by just using the digits 0 to 9 twice.

I do recall that books had random number tables.

Another source was telephone numbers from telephone books, especially the right-most digits of a phone number.  Open a page at random, close your eyes, point to a phone number at random.  If the last digit is from 1 to 6, you have a random dice throw;  if the last digit is something else, then repeat.

We used random numbers to do simulation problems then, no differently than now.

there are many ways.

use a ten sided die and repeated rolls is one. But that is not truly random as the die is not perfectly built.

or a 6 sided die numbered 0 thru 5, roll it twice, add the two, ignore any over 9. Repeat for additional digits. 

You can use some of the digits of  π, which had been calculated by hand to thousands of digits.