Once you blittet your image to the screen, it will stay there until you draw something over it. It won't disappear by itself.
An easy solution is to just clear the screen every iteration of your main loop, e.g. something like:
while running:
DISPLAYSURF.fill((0, 0, 0)) # fill screen black
for event in pygame.event.get():
# handle events
pass
# I'm using 'key.get_pressed' here because using the KEYDOWN/KEYUP event
# and a variable to track if a key is pressed totally sucks and people
# should stop doing this :-)
keys = pygame.key.get_pressed()
if keys[pygame.K_UP]:
# only draw image if K_UP is pressed.
DISPLAYSURF.blit(arrow_left, (20, SCREEN_HEIGHT/2 - 28))
pygame.display.flip()