I am working on mindfulness and mediation myself, and I still consider myself a "beginner" ;-)
I don't view it as "blocking other things out" as much as just acknowledging things for what they are and reacting appropriately. For example, if you see a raging dog running at you with it's teeth bared, an appropriate response would be to get out of the way or protect yourself immediately. Conversely, if you are annoyed by something a coworker or family member says, it's probably fine to just acknowledge it and move on ( VS. getting worked up and having a confrontation ).
Another example is specifically in my instance of health anxiety - if I have a strange feeling in a part of my body, I try to just view it as that...a strange feeling and something normal that my body does...vs. overreacting and making into some kind of possible/horrible health problem in my head.
That's where the meditation fits in for me....when you meditate you are simply trying to concentrate on one thing only. It's impossible for you to completely block out other thougths, but re focusing over and over on that ONE thing helps you learn to just acknowledge the other fleeting thoughts as just that - and refocus on your goal at hand.
I really liked the book "10% Happier" by Dan Harris, a great down to earth read on his journey through meditation and mindfulness as a beginner.