Why does Snug, who plays the lion, make a fuss about proclaiming his true identity?
Snug is one of the "rude mechanicals" in the play--the players are some of the worst actors ever, and please the audience most because of how truly horrible they are. So when he says that he doesn't wish to scare any ladies, it is funny because he has no hope of anyone actually believing he is a lion, or even of actually scaring anyone. Taken within the overall context of the play, this line adds an element of humour. It also plays thematically on the lines between reality and fantasy. Snug is comfortable with fantasy as long as everyone is clear that it is not real. He wouldn't want anyone to be hurt or offended or afraid because of the role he plays.
The mechanicals also seem to be aware of how important it is for their play to please (so that they will get paid) and not offend. The rude mechanicals may contain a veiled commentary on the lives of actors in Elizabethan society as well(it is not only Snug who is concerned with how his role is received) or, they may function as a stab at Shakespeare's contemporaries, bad actors who spend more time explaining and apologizing for their performances than on acting and entertaining.
Snug does not want to frighten the ladies (as doing so can mean death to a player) but he obviously relishes his "fierce" identity:
Here are the lines that support this claim (5.1.210-216):
(As the Lion):
You, ladies, you, whose gentle hearts do fear
The smallest monstrous mouse that creeps on floor,
May now perchance both quake and tremble here,
When lion rough in wildest rage doth roar.
Then know that I, one Snug the joiner, am
A lion-fell, nor else no lion's dam;
For, if I should as lion come in strife
Into this place, 'twere pity on my life."
(*Note: A "fell" lion is one who is dead.)