What do you mean, looking into the exact center? Obviously, sometimes we are, and sometimes we’re not. We can look in any direction at all, and obviously one of those is the correct direction, even if we couldn’t tell which way it was.
I think what you meant to ask was how we know which direction the center of the Galaxy is in. The most straightforward observation is to look at the direction and distance to the various globular clusters which orbit the Galaxy. Those aren’t restricted to the disk, and form more or less a sphere. And other spiral galaxies we can observe have the center of their globular clusters and the center of their spiral in the same place, so we assume that’s true of our Galaxy, too.
Then, once you’ve got that, you’ll notice that there’s an extremely strong radio source in that direction, stronger even than the Sun in those wavelengths. This object, called SagA*, appears to be a supermassive black hole, much like the ones we observe in other galaxies, with the radio emission coming from the accretion disk around it.
And further, if you look at the stars in the immediate vicinity of SagA* (you have to do this in infrared so you can see through the dust I mentioned earlier), you can actually see those stars orbiting around some invisible object. From measurements of the orbits, you can determine that the thing they’re orbiting around has a mass of 2.6 million times the mass of the Sun. And a black hole is the only thing that can be both that massive and that small.
Myself, I’d look for the plane of the galaxy, AKA the milky way, as it is also called, and notice how it gets much denser in one region, and least dense at about 180° opposite that in the sky.
The plane gives me a fix in one dimension.
The density gives me a fix in the other dimension.
Only a few globular clusters are visible to the mk 1 eyeball.
Yes, but I think that supergalactic’s point was that the denser region might just be us looking along the spine of our spiral arm, which would not be the same direction as the actual core.