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.