I watched a Synthesia video of this music paper, and on the Synthesia is played that note as a A sharp, but I see no sign of playing an A sharp.
It's not an A♯, it's a B♭. The key signature tells you that all B's you come across are flat hence this B is flat unless otherwise stated.
See the related question: Is a high A in the key of D flat still flat?
The note in the music is a B note.Of some sort, not an A of any sort! Count up, and that line will be a B. As Dom says, because of the key signature of one flat, which happens to be the note B, then that note is played as B♭.
Whilst A♯ and B♭ are the same black key on the piano, they're not always the same note on other instruments - but that's for another day! And in any case, when there's a B♭ in the key signature, it's extremely rare that the same sounding note will be named (or written) as A♯.
As others have explained, that note is B♭, not A♯. The B♭ in the key signature applies to all B notes - regardless of position on the stave.
It may help to think of the scales with a "each letter must appear only once" rule. So, for the key of F only one of the following is correct:
a) F, G, A, A♯, C, D, E, F b) F, G, A, B♭, C, D, E, F
It is, of course, b) as each letter appears only once.
If we apply this rule to a more awkward key such as F♯ we get:
F♯, G♯, A♯, B, C♯, D♯, E♯, F♯
Notice here that we have to write E♯ to comply with the "each letter must appear only once" rule even though it would be simpler to write 'F'.
This is a B flat. Technically, every flat is the sharp of is preceding note if it is not a natural half tone gap, at least within the same temperament. Thus, B flat is the same as A sharp.
Also, no instrument I am aware of has distinct keys or strings for A sharp and B flat.