To add on to what Rogers said, double flats are the same no matter what key you're in.So even if a flat is in the key signature, you don't double the flat on the flat. Meaning in B flat minor (the key in your example), it says B double flat, you would play that as an A natural because you're double flatting the B natural, not the B flat.Rule of thumb with double flats and sharps: ignore the key signature, and double flat or sharp whichever note the composer asks for.Does any of that make sense? It can take awhile to get used to.
Double Flat means, lower the note by a "tone". Here's some listing (All double-flatted):C = B-flat --- F = E-Flat ------ D# (E-Flat) = C# (D-Flat)D = C --- G = F ----------- F#(G-Flat) = EE = D --- A = G ------ G# (A-Flat) = F# (G-Flat)B = A --- C# = B ------- A#(Commonly known as B-Flat) = A-flat(G#)