This is so you actually do check the door notifications rather than simply pushing through when in a hurry or assuming you have everything you need and have checked all relevant notifications when in a hurry, and could also allow syncing with multiple devices.
The benefits of a watch are that it's easier and quicker to check than a phone and easier to feel it if it vibrates so you don't even have to take it out of your pocket.
The smart door display would be easier still, but here you would also benefit from the haptic feedback from the watch before you leave the house, if there are important notifications to check.
The combination of the instant feedback from the phone or stronger vibration from the watch, while you are at the door, means it's quick and easy to glance at the door rather than digging for a phone or even waiting for your watch to wake up and potentially scrolling in either case.
You are therefore less tempted to stop when in a hurry to ignore notifications if you have too many, and making that assumption that you already know what the notification is for rather than risk being late or missing a bus by unlocking your phone and scrolling through notifications that may be less important before you can even see the specific text.
For an example of this multi-device sync, aside from the watch also sharing your phone's location-based reminders, as @Samuel Bello mentioned, this could go well with other smart devices and wearables like the weighted bag, if it was a smart bag that contained highly sensitive electronic sensors to also help you know if you've forgotten your keys, wallet and other items.
This would work potentially similarly to the way sensors in the smart water bottle by Bellabeat track your water consumption and sync to an app).
This way if it is your watch or bag or phone that is forgotten you also have reduncancy in the notifications (since your phone can't alert you if you've forgotten to put it in your pocket!)
In the case of forgetting something, the watch could then make you aware through special vibrations which so correspond to text on the door.
This could apply to other crucial notifications or reminders you might not have been able to pre-schedule yourself (and therefore think it safe to ignore), such as new information like Google sending you a severe weather warning when you're about to drive to work, or display the text on the door of a coworker from a specific work group sending you a request to bring something for a presentation.
More crucial and urgent or information such as the weather warnings could give extra bright/animated text on your door. Given potentially multiple notifications compete for space in the attention economy, having these different ways of prioritising more important/urgent types of notifications into the type that would show up on your door and get a particularly strong watch vibration or loud warning siren from your house if ignored, for example, will become more and more relevant in dealing with big data and determining which notifications are most crucial for us to adhere to first.
Potentially, your door screen could also animate that you are receiving a phonecall from a prioritised number, so if your relative is calling you regarding an emergency and you were just about to walk out to the store without your phone or had your phone on silent or off, you will DEFINITELY get notified if you are at home, or else see a prioritised/animated missed call/messages notification when you do reach the door if you somehow slept through it or got home late, rather than relying on your phone and/or smart watch which may be off/ on silent /charging and would be less obvious than brightly animated door text.
A tool like this would be particularly useful for a person who is partially sighted and struggles to see smaller screens or who is deaf or hard-of-hearing and would benefit from more visual notifications rather than sound to ensure they didn't miss something crucial if they do not have their phone in their pocket while at home to feel the vibration.