Production of yeast

Almost certainly the starter yeast is yeast slurry that's been stored frozen in liquid nitrogen. Interestingly, one of the most common methods is to store it inside sealed-off portions of plastic drinking straws.

Commercial yeast labs have large collections (sometimes called libraries) of pure cultures of different strains of yeast stored this way (pure culture meaning that all cells in the culture are genetically identical). These "stock" cultures, as they're called, are essentially used to seed larger and larger fermentations, in which the yeast cells multiply, until they yield enough yeast to divide up into "pitchable" quantities. Based on how brewing yeast reproduce (by asexual budding) all growth (new yeast cells) should be genetically the same as the mother cell, though of course mutations do occur.

These pure cultures, originally, are either:

1. Acquired from another laboratory/collection OR
2. Isolated from colonies cultured on media (agar plates), which can generally be ensured to have grown from a single cell (sourced from other beers, wild fruit, really anywhere you might find yeast).

There are of course many other methods for storing yeast long(er) term, such as sub-culturing regularly, freeze-drying or vacuum-drying (instant/active-dry yeast), but liquid nitrogen is generally considered the most genetically stable (and is relatively straightforward).

You absolutely can do this kind of stuff at home, it really is only limited by how far you want to take it. In general at home you'll simply be making regular sub-cultures, either in liquid media or on agar slants, however there's nothing (aside from experience and know-how [and money]) keeping the home brewer from succeeding using the exact same methods as commercial labs do.