This is a couple years old, and the querent has probably long since graduated the classes they were taking. But I used to have some trouble with this, and all the "this equation tells us it's true" approaches didn't really help conceptualize it, so maybe my approach will help someone else with a similar question in the future.
The way I always thought about it was a merry-go-round. At first, it's stationary. You push with some force, the merry-go-round starts spinning. So now it's moving a bit, you grab the next bar on the edge and push some more, and it starts spinning faster. Eventually, it's spinning quite fast.
So now, in order to spin the merry-go-round more, your hand has to rapidly accelerate to catch up to the moving bar, then when you make contact you can start accelerating the merry-go-round some more. But most of your force was used up trying to accelerate your hand, so you can't get much force on the merry-go-round.
Eventually, the merry-go-round is moving so fast that your hand accelerates until it's just barely touching the bar, but can't actually push on it.
The moral: the faster something is moving, the less force you can put on it because a lot of the force is wasted on your hand to catch up with the moving object.
In our case, we're saying force on the object is constant no matter it's speed. As it goes faster, that means there's constant force on the object, plus ever-increasing force just to catch up to the object to start applying that force. More force requires more power.
The same thing happens with a motor that spins gears to accelerate an object via tires or chains, etc. As you go faster, you need to change gear ratios to keep the engine at a speed it can handle. Lower ratios mean the engine speed goes down, so you can go faster, but it also reduces the torque multiplication, so you can't accelerate as fast at that higher speed.
In order to maintain constant acceleration, you need to produce more torque at higher speeds to make up for the gear reduction. More torque requires more power.
These examples involve a stationary object accelerating a moving object. What about a rocket or something? Well, to accelerate this way, we need to launch mass out the back of the rocket at high speed. If we keep launching it at the same rate (mass per second) and speed (meters per second), we get constant acceleration. But there's a problem. We also have to accelerate the reaction mass we're using to accelerate.
So to launch a bunch of mass out the back at some speed, it takes some amount of energy. But to launch the same mass out the back at the same speed once we've already started going, we have to first accelerate that mass to our present speed (if we didn't accelerate it, it wouldn't be in our rocket, so we couldn't use it to go faster). So we spend some amount of energy getting it to our present speed before spending more energy launching it out the back.
The faster we go, the more energy it takes to get the next bit of reaction mass to the current speed, just so we can throw it out the back to go even faster.
In all cases, constant acceleration requires constant force, which requires more and more energy per second as we increase our speed. More energy spent per second requires more power.