Why did the United States support that Panamanian rebels and recognize the Republic of Panama?
The basic reason for which the United States supported the Panamanian rebels was the US's desire to build a canal across the Isthmus of Panama.
The US had been negotiating with Colombia for the right to build this canal. However, the Colombian Senate overwhelmingly rejected the treaty that would have given the US this right. The Colombians felt the treaty did not give them enough in return for their loss of control over their own territory.
Unhappy with this outcome, President Roosevelt moved to support the Panamanian rebels. Once they had thrown off Colombian rules, the US made a somewhat suspect treaty with Panama giving the US the right to build the canal.