What are the number of output bits of the internal function F used in BPS algorithm? In the BPS specification document it says


  We denote by f the number of output bits of the internal function F


What is this if the internal function is AES? Would it be the block size (e.g. 128)?
Yes, in BPS, if the internal function $F$ is a block cipher, then $f$ is its block width, thus 128 for AES (for all key width variants). That's clear enough in the paragraph quoted in the question:


  We denote by $f$ the number of output bits of the internal function $F$. (...)
  If $F$ is a $f$-bit block cipher (...)
  We denote by $F_K(x)$ the application of the block cipher $E$ with the key $K$ on the plaintext $x$ ($F_K(x)=E_K(x)$), (...)