Because the constitution specifically gives the power of impeachment to the house of representatives.
What a lot of people get mixed up about, is the fact that impeachment does NOT remove the president from office.
Impeachment - which is done by the house - is the process of formally bringing charges against the president.
AFTER the house impeaches a president, the Senate holds a trial to evaluate whether or not the president is guilty of the charges and whether or not the offense is severe enough to remove the president from office.
So yes, the house has the power to impeach, but that is a relatively meaningless unless the senate follows through to remove the president.
The most recent example of this: Bill Clinton was impeached by the house for lying to congress during the investigation into his affair with Monica Lewinsky, but he was never removed from office by the senate, so he served out the entire 2nd term of his presidency.
Also something that some people don't seem to understand - if the president is impeached and removed from office, the vice president becomes the president. It does not revert to the 2nd place finisher in the election. There are some people who seem to think a Trump impeachment (and removal from office) would make Hillary Clinton the president. That simply is not the case. Mike Pence would become president.