How does diethyl ether behave like a base?

There are a variety of different definitions of bases: 
the most limited is that acids produce H+ ions in solution while bases produce OH- ions
in solution.

A somewhat broader definition is that acids
are proton donors while bases are proton acceptors. This allows for amines (CH3NH2) to
act as bases.

The broadest definition is that an acid is an
electron pair acceptor while a base is an electron pair donor.  It is in this context
that ethers can be considered bases.  An ether has an oxygen atom in the center with two
organic groups attached. In your case, diethyl ether
is:

C2H5 - O - C2H5.  Because the oxygen has two electron
pairs it can act as an electron pair donor.