You could use something like this:

awk -F, 'NR == FNR { a[$0]; next } $2 in a' A.txt B.txt

This saves each line from A.txt as a key in the array a and then prints any lines from B.txt whose second field is in the array.

NR == FNR is the standard way to target the first file passed to awk, as NR (the total record number) is only equal to FNR (the record number for the current file) for the first file. next skips to the next record so the $2 in a part is never reached until the second file.

alternative with join

if the files are both sorted on the joined field

$ join -t, -1 1 -2 2 -o2.1,2.2 file1 file2

00,aa
11,bb

set delimiter to comma, join first field from first file with second field from second file, output fields swapped. If not sorted you need to sort them first, but then awk might be a better choice.

There seem to be two schools of thought on the matter. Some prefer to use the BEGIN-based idiom, and others the FNR-based idiom.

Here's the essence of the former:

awk -v infile=INFILE '
  BEGIN { while( (getline < infile)>0 ) { .... } }
  ... '

For the latter, just search for:

awk 'FNR==NR'