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.