Answer and Explanation:
The speaker does not want to be "somebody", because being somebody imposes a dependency, in making herself noticed. This dependence is tiring, repetitive and takes away the individuality of the individuals, because they make that, to supply the desire to be someone, they have to repeat the same actions, actions again and again. It makes him a follower, a dependent and she doesn't want it for her.
The lines that best explain this feeling are:
"How dreary - to be - Somebody!
How public - like a Frog -
To tell one's name - the livelong June -
To an admiring Bog!"