How did the United States gain the control over the Panama Canal? What really happened in the negotiations between the U.S and the Colombian...

First of all, please note that when the US and the
Colombian government were negotiating, the canal did not exist.  The US got access to
Panama through the negotiations and then it was the US that built the
canal.

In 1903, the US and Colombia concluded a treaty that
would have given the US the right to build a canal through the Isthmus of Panama (this
was part of Colombia at the time).  But the Colombian Senate rejected the treaty.  This
made Pres. Theodore Roosevelt very unhappy and so he helped Panama stage a rebellion
against Colombia.  The US Navy prevented Colombian troops from crushing the rebellion
and Panama became independent.

After that happened, the US
signed a treaty with the Prime Minister of Panama (who was actually a Frenchman who was
part of a company that wanted to make money off the canal).  The treaty gave the US the
canal zone.  After that, America built the canal, completing it in 1914 after 10 years
of work.