Where are you seeing that it "says high demand"? The only place where demand is indicated is on the route planning pop-ups, and if supply (the green line) is more than demand (the left-most blue line) then it's full.
Do you mean that multi-colored line on the Open Routes menu, the one that goes from red to green that says "Passengers"? That simply indicates the relative size of the airport in terms of pax, it's not necessarily indicative of demand on that route. It's usually a decent barometer, but smaller airports could have larger demand to your airport or vice versa.
As for the demand graphs, the furthest left bar indicates the total number of people that want to fly it. The green line in the middle indicates the total seats being offered already. Then there's another blue bar that indicates the number of seats you offer. If the green line is higher than the left line, those seats are flying empty (generally, there are exceptions, but I'm simplifying). You want to minimize the height of the green line above the demand unless you're purposefully trying to force competition off the route, which I don't recommend at your stage.
In your case, that route is fairly full. And your competitor is a huge airline, meaning the passengers are going to want to fly him a lot more. You can either keep making that flight and try to steal the passengers away, or you can try some other route. The benefit to keeping more flights on the same route, is that there are fewer costs. You've already got ground crews and other personnel to that destination. Additionally, each destination increases your necessary marketing spend. So you can afford to fly a bit emptier because you've got less costs to cover than if you seek out a new route. On the other hand, it will be harder to steal seats from KidCo, so you'll need to reduce your ticket costs. You also run the risk of drawing attention to yourself by trying to dominate a route, and when that happens KidCo might decide to cut his rates to nothing and run you out of business; which he can do on a whim if he so desires. Personally I would hesitate to put at least a couple more flights on that route, as there's enough demand there to keep another couple flights running pretty full.