Let me see if I can. Before we go to space, let's try something on the ground. Imagine pitching a ball horizontally. What do you expect if you pitch it too slow? The ball will curve more towards the ground and meet it early, won't it? (In other words, it doesn't go very far and doesn't stay airborne for long). Going from ground to space, this action remains the same. You need to 'lower an orbit'? Reduce its forward velocity. It will curve more towards the planet and reach closer to the ground.
However, there is a bit more detail involved here. Why doesn't the satellite just fall to the Earth? (Please excuse me and disregard this part if you know this already. I'm trying to maintain conceptual continuity.) So, when something is flying horizontally (no aerodynamic forces), we know that its trajectory will curve towards the Earth due to the pull of gravity. If the ground (on Earth) curves as fast as, or even faster than the trajectory's curve, the object will never get an opportunity to even reach the ground. This is 'orbiting'.
Now assume that the satellite is initially in a circular orbit. The gravitational force acting on the satellite at any point in the orbit is perpendicular to the satellite's velocity vector and tangential to the orbit. The satellite will maintain a constant speed at this point, since its velocity and the force are always perpendicular [1]. So, what happens when we reduce the satellite's forward velocity? Just as we've seen with the ball, the satellite's trajectory (orbit) starts to curve more towards Earth. Now a subtle, but important change occurs. The velocity and the gravitational pull are no longer perpendicular! They start to align! And when that happens, the speed MUST increase. So, the satellite is now losing altitude and speeding up simultaneously [2]. At some point, the satellite will pick up enough speed again to 'straighten its curve' and avoid falling to the ground. In effect, the satellite had to compensate for the lost velocity in order to remain in orbit, and it did so by exchanging some of its altitude (gravitational potential energy) for velocity (kinetic energy) [3].
So our satellite 'fell' from where we slowed it down, until it had enough velocity again to maintain orbit. At that point, the gravity and the velocity are parallel again, since it will keep falling otherwise [4]. But since it 'fell from a higher altitude', it's speed is now too high for it to remain at that altitude. The orbital curvature is a bit 'too straight' now and it starts to curve away from Earth. So now we're in the exact opposite situation of what was explained in the last paragraph. The satellite is now climbing back up again! As it happens, the satellite actually climbs back up to the point where we slowed it down! And when at that point, its velocity is exactly the same as what it was, after we had slowed it down! [5] So the satellite did the inverse of what it did earlier - it exchanged kinetic energy to get back its altitude (potential energy). The satellite is now living in cycles juggling kinetic energy and potential energy back and forth. The final effect is that the point in orbit that's diametrically opposite to where you slowed it down, is now at a lower altitude. And thus you've effectively 'reduced the orbit'!
One more detail to pin down. How do we slow down a satellite in the first place? Easy! Push the satellite in the opposite direction of its velocity [6]. This is called 'retrograde thrusting' or 'retro burn'. But that's about as easy as it gets. Remember that unlike on Earth, you don't have a surface (a wall or the ground) to lean against. Imagine pushing something heavy on an ice rink. The good news is that you can still push things on an ice rink. The only catch is that the push force will set both the item and you in motion in opposite directions [7]. And that's exactly what we do in space. We throw out mass from the satellite in the form of super-fast gaseous of plasma exhaust. The key is to throw out the mass with as much momentum as possible. But the mass is limited by how much you can carry - it's a depleting resource. So you're basically left figuring out how to throw it out with ever increasing speeds. And that's how we slow down the satellite in space - fire your thrusters!
And finally to lower an orbit entirely, instead of just one point on it, you have to do multiple firings. There are bunch of these 'orbital maneuvers'. The most common one is the Hohmann Transfer [8]. If you could understand what's given above, most orbital maneuvers including Hohmann Transfer will feel very intuitive to you.
[1] Speed is the magnitude of velocity and it remains steady in a circular orbit. However, the perpendicular force will keep bending the velocity vector, thus constantly changing its direction.
[2] This is the from-the-first-principles explanation of conservation of angular momentum. This is how the ballerina spins faster by pulling in her arms.
[3] If this sounds like a 'negative feedback' phenomenon to you, that's because it is. Feedback is a mathematical construct. Nobody ever said that a feedback mechanism must be implemented separately. Some systems have them inherently built-in.
[4] This is the lowest point of the orbit - the periapsis.
[5] Yes. There is quite a bit of hand waving here. I didn't explain why the satellite went back to its original position with the exact same speed. But that's what actually happens. It might take a lot more 'mathematical sense' to explain just using words. One thing I know is that this has something to do with the fact that the gravitational field is one of those 'conservative fields'. If you take a trip inside a conservative field, and return to the location where you started, you will be left with the exact same (kinetic) energy as you started with. You may exchange your energy during the trip, but you always regain it back when you get back to the starting point, no matter what path you took. As far as I understand, the 'conservative' part refers to the part that the energy is conserved and stored, and never lost. Unfortunately, the force field that we're most familiar with - frictional force - isn't conservative at all. If you're going on a trip, be ready to spend some energy!
[6] One matter that confuses a lot of people is why the satellite's position changed at the opposite side of the orbit, instead of the point where we applied the force. The answer is in the Newton's second law. Force changes momentum, not position - at least not directly. The direct effect of application of retro thrust is that the velocity reduces at that point. The change of position on the other side of the orbit is only a consequence of that velocity change.
[7] Yes, the Newton's vengeance law.
[8] https://en.wikipedia.org/wiki/Hohmann_transfer_orbit
[9] Every so often, someone comes along and argues that gravity is not a real force and all these explanations are wrong. If you want to deal with this in terms of relativity and space time curvature, be my guest. But for all practical purposes, the old faithful Newtonian physics works just fine, even as a special case of relativity.
[10] This should probably have been a blog post. Please don't shout at me if it annoys you. This is one of my favorite subjects and I just got carried away. I used to teach and train many students and junior professionals in these topics.