Your arms can be a really valuable asset when combined with pumping your legs to generate speed. If you watch Kelly Slater’s recent 10 point ride at the Pipeline Competition you will see how he used his arms and legs to build speed and make it through 3 sections whilst being barrelled the whole way.
Martin Dunn has excellent learn-to-surf videos and he talks about using a catch phrase eg "bend and throw". The term "pump and lift" also applies as you need to pump your legs first then lift your arms simultaneously.
Your arms can also be used to assist you when you need to get your board up the wave and either hit the lip for a re-entry or travel along the lip with a floater. Aerial manoeuvres also require a great deal of arm movement to assist you with getting the board in the air.
Really good surfers use their arms cleverly in critical situations eg they can be late with their bottom turn with the lip already on the way down but still make the turn and keep going. Crucially they use their arms to lift into the lip and then return down the face of the wave.