I've gotten confirmation from the community manager that hand-eye does in fact help with batting down pucks in the air. So, there's that.
Offensive awareness has to do with how well you react to passes from teammates, like if you're trying to take a one-timer. It also affects your ability to pickup loose pucks. So, say you're driving the net and your opponent knocks the puck away. Having a high offensive awareness will help you pick the puck back up immediately.
Defensive awareness is pretty straight forward. Helps you intercept passes, disrupt plays, etc.
Strength is an interesting one. If you just go by the stat description, strength supposedly just helps you stay on your feet when you collide with another player and helps you control your opponent during net battles and boardplay. But, go look at the zone/superstar abilities and read the requirements for them. Most of the abilities' required stats are related to what that ability specializes in. Tape to tape? Passing and offensive awareness. Shutdown? Stick checking and body checking. One tee? Hand-eye and slap shot power.
So, take a look at abilities like Bouncer and Total Eclipse. Both of those abilities are supposed to help you in the gritty situations, i.e going to the front of the net to screen the goalie (Total Eclipse) and defending your own net front (Bouncer.) It seems to me like the strength stat helps you with not getting bullied, and also your ability to physically overpower other players.
Now, there is another ability that requires strength, Thunderclap, but I'm not exactly sure how strength ties into slapshots, which is what Thunderclap helps. The Bouncer description does mention stick tension and limb strength, so my guess is that the strength stat aids those things and is thus related to things like slapshots.
To be clear, this is just my own particular guesstimation and fan fiction, so I can't guarantee I'm correct about strength. I can only infer based on EA's descriptions.