Firstly, the galaxy is only about 1000ly thick. We are fairly close to the galactic plane, maybe around 65 ly 'above' it if we call the direction we are moving away from 'down'. So on your assumption that visible objects are all within about 1000ly, we can suppose that we should see more stars in the plane than above and below the plane, as there is only about 500ly of stars in the galaxy above and below.
We are perhaps 25kly from the galactic centre. Therefore there are about 25kly of stars outwards, and 75kly of stars inwards. If we could see all of them with the naked eye, we would see more stars to one side than the other. While many of the stars that we see are within 1000ly, that doesn't mean that the stars within this region of visibility are evenly distributed. Stars are packed closer together nearer the centre, which implies that we will see more towards one side than the other.
I don't know where you get the idea you seek confirmation for that the density of stars is not higher in the galactic plane, because it isn't true. Density is greater toward the centre of the galaxy; and orthogonal to the galactic plane, density decreases as you move away from it.
Plus, we see far more than merely dots of light from other stars. We see other objects that are aggregations of stars - clusters, and galaxies like Andromeda, the LMC and the SMC. We see nebulae, clouds of gas and dust. In fact, dust even obscures what we might otherwise see toward the galactic centre.
So when you ask 'what is it we see?' the answer is: lots of things. Because there is more of that 'stuff' in the plane of the galaxy, we see the majority of that stuff as a ring around us. Because there is more of that stuff towards the centre of the galaxy, we see that the ring is brighter, fuller, more complex, and dominant on one side.