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This is not ICE. CBP is Customs and Border Protection. They deal directly with the border and other entry points like airports, border crossing and ports. The easiest example of what they do is to point you towards the movie The Terminal. In that movie, Tom Hanks plays an Eastern European, Viktor Navorski (fictional country) who gets denied entry to the U.S. upon arrival when a government coup sees his visa invalidated mid flight. The film depicts head of CBP at JFK airport in New York, in a struggling battle to keep Hank’s character in the international terminal, the only legal area he can occupy after determining that they can’t send him back amid a civil war which the U.S. has yet to recognize a formal government. The conflict directly centered around the recognition of his country’s government. While in mid flight to the U.S. the civil war began and the government was dissolved. Upon landing at JFK, he no longer had valid paperwork to enter the U.S. After determining he could not enter the country legally, it is made frighteningly clear that without a government that the U.S. recognizes, he could not legally be returned home either. The movie depicts Viktor’s struggle to continue his life inside the international thermal of JFK, awaiting the day he can finally leave.

Immigration and Customs Enforcement deal with immigration and customs enforcement. So this organization works inside the borders of the country. They deal with immigration and customs violations, and trafficking and cross-border crimes.

These are not the same organizations and they serve different purposes. Both fall under the authority of the Department of Homeland Security. People are conflating the two organizations since they deal with the legality of being in the United States but they have different methods of operation.

EDIT: CBP try to stop you from entering the country. ICE try to eject you once you have gotten inside.