SQL Server linked server performance tuning

The query plan should tell you.  

I don't see how SQL could that for a query against a remote db source.  I would expect it to re-read the remote data every time.

Within the same query execution, yes. Otherwise no. MSSQL has no means to reliably determine whether content has changed on the other side, so caching would be very, very difficult.
If you need to access the (unchanged) data very frequently, it is a good idea to build a local copy of the table(s) once a day.

Thanks for the responses but perhaps I shouldn't have suggested my guesses since I have no idea.  My premise is that I have a non-trivial amount of data flowing from a ODBC connection via linked server thru  SQL linked server (somehow? and I want to have a deeper understanding of how SQL is managing it. Armed with that knowledge, I am hoping to be able to   is to understand Let me restate the base question:

What are the internals of how a linked server query is processed?

The query optimizer tries to get some meta data first (like data types), and tries to determine if it can send some filter data over to the ODBC data source. In most cases it can't, so has to read all data, and then process them locally. For that, tempdb and in-memory DB is used.
For that reason it is always a good idea to use constructs like

select one, two, three
from localtable lt
join (select a, b, c from remotesrv.remotedb..remotetable where bookdate between '20150101' and '20150201') rt
on lt.key1=rt.a

Select allOpen in new window

In general, the correct assumption is "remote query cannot be optimized", so main target should be to restrict the amount of data selected remotely in the first place.