Why is it bad if air gets into the blood?
Because it can kill you if you get enough, causes an air embolism and can stop the heart.
Air bubbles can potentially block little arteries and affect the organ the artery supplies, for one thing. What it does depends on where it goes, but air bubbles to the heart, the lungs or the brain could be very serious.
In practice, there should be no air bubbles going into you during a treatment. Dialysis machines are designed to prevent this. In my experience so far, the biggest and most constant risk of air bubbles is from connection of the venous bloodline to the venous needle line (the arterial needle line and bloodline are not as critical, because it’s taking blood from you rather than putting it in you, and this blood will go through the machine’s safeguards before coming back into you). Luckily, little bubbles there, introduced when making the connection, probably aren’t going to be harmful, but still, the connection has to be done very carefully using proper technique in order to minimize this.
For a person doing his/her own dialysis, it’s extremely important to inspect the venous bloodline very closely to ensure all the setup went well and there are no air bubbles in the line before connecting and starting the treatment. This could happen if a clamp were left open or something when it shouldn’t have.
Air bubbles can also be introduced when using a catheter, but my only experience is with a fistula.