When you use your back, you engage the core of your body, which implies making a maximum effort to do or achieve something as quickly and effectively as possible. The idiom 'put your back into it,' therefore, is used to express an urgent need to get something done, to refer to doing something, especially physical work, with a lot of effort, or to encourage somebody else to do so. It is so popular as a motivational phrase that it has become something of a cliché. As the video shows, it can be used informally in both a social and professional context. Phrases with a similar meaning include 'go all-out,' 'give it everything,' and 'a bit of elbow grease.'