These "arms" are actually proteins that sit in the outer membrane of the virus. There are three viral proteins in the outer membrane of influenza: M2, HA, and NA; however, M2 does not protrude significantly outside the membrane, so the proteins that you are see in the electron micrograph are HA and NA.
HA, short for
hemagglutinin, is a protein that allows influenza to bind to mammalian cells in order to infect them. It is also involved in fusing the viral membrane with the cell's membrane, allowing the virus's genome to enter the cell and begin replicating new viruses.
NA, short for
neuraminidase, is an enzyme involved in clipping sugar molecules off of the surface of cells. Since these surface molecules tether budding virions onto the host cells, NA is essential for the release of new virions into the body so that they can infect other cells. Notably, the drug tamiflu targets NA.
For completeness, I'll note that
M2 is an ion channel that allows the virus to sense the pH of its surroundings. This is important because once the virus enters the cell, it is in an endocytotic vesicle. Usually the cell will try to digest material that it takes up through endocytosis, and the first step in this process is acidifying the interior of the vesicle. M2 conveys this acidification to protein inside the virion, triggering the mechanisms in the virus that cause the release of its genetic material into the cell.
