Radio image of a quasar (MPIfR), the tiny dot in the middle is the actual black hole, and the flame-looking signatures are the radio jets protruding several light years from the core
To be able to see an object that is this far away, we rely on the object to be a strong source of radiation. For the VLBI measurements, quasars (quasi stellar objects) are used due to their very intense radio wave emissions. A quasar can be described as a black hole, with an accretion disk (where matter is drawn into the gravity field of the black hole). Due to the immense pressure at the center, the quasar emits violent jets of radio waves perpendicular to the accretion disk. These radio jets can protrude several light years in each direction and these are the radiation signatures that we observe with our telescopes.
Very briefly explained, what we do in VLBI is to measure the time difference between the arrival of the signal at different antennas, and by doing so we can calculate the distance between those antennas
The VLBI technique
The earth spins and it moves through space, the continental plates drift, and basically nothing stands completely still. This makes it hard to find a stable reference point for positional measurements. One solution to this problem is to use stars that are very far away (routinely they are more than a billion light years distant) , even if they too will move, it will seem as if they are stable. At least stable enough that we can use them for positional calibration. By repeatedly and continuously updating our knowledge of these distant objects we keep up our knowledge of exactly where we are.