"to beat out" refers to repeated application of force (usually with a hammer or some other blunt instrument) to, as others have noted, push the material outward.
The contrast I think they are trying to make here between "beaten out" and "pressed down" is that there are generally two ways of forming a rivet into a flat surface (or making anything flat):
- by consistently applying pressure on the rivet while its molten (pressing down, usually with some sort of mechanized press), or
- by repeatedly applying short bursts of pressure (beaten out)
Note this also happens to other things: clothing, for instance, can be "pressed" or "beaten" to get out wrinkles (though not many beat laundry nowadays). Containers (such as walls or doors) are often noted to be "pressed upon" or "beaten against". Even people, when in a bad state, can be "pressed" (referring to being stressed out, etc.) or "beaten down" (usually had a confrontation or bad day, etc.).
I think the key here is to not worry to much about the preposition or other word that comes after a verb, but think of it as part of the verb to remember (in context, unfortunately). This is a common approach of linguistics, as well (Distributed Morphology, HPSG, Construction Grammars, etc.).