When you do something blatantly, you do it obviously, without trying to hide it. You might blatantly lie to your boss if you don't care about getting in trouble.
Something done or said blatantly is bold and shameless. You might be doing something bad, but if you're doing it blatantly you're not worried about repercussions. In 1596, the poet Edmund Spenser invented the word blatant, using it to describe a slanderous beast. In the 1600s blatantly meant offensively loud, and by the late 1800s it had its current meaning of "conspicuously, obviously." A similar word is flagrantly, but you'd use it to mean "shockingly," while blatantly is closer to "obnoxiously."