You place silver in a closed oven and heat it until the silver evaporates. You now have a box of silver atoms going around in all directions. You open a small hole in the oven and pointspoint it towards your experiment. Now only silver atoms going in that direction can escape the oven, thus giving you a stream of atoms in that general direction.
By setting up more walls with small holes in them, you can sort away any atoms which are not going in the completely right direction, depending on how well-defined you want your beam to be.
See e.g. this reference.