How did changes in voting procedures enable government leaders to be elected more democratically?

I assume that you are talking about the changes in voting
procedures that happened during the Progressive Era.  During this time, there were a few
changes, mainly at the state and local level, that allowed more democratic
elections.

The most important of these was the secret
ballot.  In many cities before the Progressive Era, secret ballots were not used.  The
use of secret ballots was meant to help break the power of urban political machines and
allow more democratic elections of city officials.  This was also the reason for the
institution of non-partisan and at-large city elections.

On
the state level, many states started using primary elections during this time.  This
allowed people to actually vote to decide who the candidates for each party would be
instead of having party bosses make the choices.  This was an important step towards
democracy.  Finally, the 17th Amendment required US Senators to be directly elected by
the people instead of being selected by state
legislatures.

All of these changes in voting procedures
increased the amount of democracy in the US.