Show and explain the reaction of people around the prison in chapter one? And why does the novel begin with prison gates?

Part of the key to your question comes at the beginning of
the chapter:

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The founders of a new colony, whatever Utopia of
human virtue and happiness they might originally project, have invariably recognised it
among their earliest practical necessities to allot a portion of the virgin soil as a
cemetery, and another portion as the site of a
prison.

No matter what hopes
the founders of our country had, even  the most optimistic about its possibilities, the
had to realize that two of the not so optimistic realities of human life, crime and
death, had to be acknowledged and provided for.   Its description fits its purpose, and
specific mention is made to the "iron" work, reminiscient of the reference made later to
Dimmesdale's need for an "iron framework" (faith) which, while it confines him, is
necessary for his support.

Hawthorne notes that in their
society "religion and law were almost identical."  The Puritan code of ethics was very
strick, for practical as well as religious reasons, and Hawthorne talks about this when
he speaks of the women who came to watch Hester's
punishment.

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Morally, as well as materially, there was a
coarser fibre in those wives and maidens of old English birth and breeding than in their
fair descendants, separated from them by a series of six or seven generations; for,
throughout that chain of ancestry, every successive mother had transmitted to her child
a fainter bloom, a more delicate and briefer beauty, and a slighter physical frame, if
not character of less force and solidity than her
own.

Things that we now take
for granted  because we have "less force and solidity," the took much more seriously. 
Their reaction is based on the strictness of their personal beliefs, but it's also
interesting to note that Hawthorne's description of them seems to put them out of the
"range" of Hester's sin:

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The age had not so much refinement, that any
sense of impropriety restrained the wearers of petticoat and farthingale from stepping
forth into the public ways, and wedging their not unsubstantial
persons, if occasion were, into the throng nearest to the
scaffold at an execution.  [bold/italics
mine]

It's easy to judge
others for sins that we can no longer see ourself committing; in fact, the most
sympathic among the women are the young who, perhaps, have only too familiar
acquaintance with Hester's sin.

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Ah, but,” interposed, more softly, a young wife,
holding a child by the hand, “let her cover the mark as she will, the pang of it will be
always in her heart.