Means plain old hard graft, usually scrubbing or polishing something that will not magically fix itself. If someone says it needs a bit of elbow grease, they mean stop looking for shortcuts and put some muscle into it. Proper everyday phrase, especially for cleaning jobs that are grim but doable.
"This oven’s minging. Nan’s coming round in an hour. Grab the Fairy liquid and give it some elbow grease, mate, or we’re getting absolutely rinsed."