Why was the US so poorly prepared for the Korean War

First, the period from the end of WWII until the outset of the Korean War was one of drastic military spending cuts in the US. The US Military was cut something like 90% from 1945 to 1946. By 1950, the US military was a shell of its WWII glory. The thinking of the time was that the US had a monopoly (until 1949) or numerical superiority on nuclear weapons, so a massive standing army wasn't necessary.

Second, MacArthur didn't care about Korea. He was ensconced in his office in the Daiichi building in Tokyo, busy building his legacy as the conquerer and rebuilder of Japan. When supplies were sent to East Asia, he took the best for his forces in Japan and sent the scraps out to Korea. The best weapons, the best officers, everything stayed in Japan.

Third, the Korean People's Army (KPA) was filled with veterans - Koreans who fought for the People's Liberation Army against the Kumingtang during the Chinese Civil War. Kim Il-sung requested that Mao send the Korean veterans back to him, which Mao did.

So on June 25, the North attacked the South with a force that was superior in every way - numbers, supplies, talent, motivation. The South Korean forces were armed with leftover Japanese weapons, many of them pressed into service against their wishes, many of whom didn't care about the politics of the day and/or were sympathetic to the Northern/Chinese communist ideals. Most of them broke and ran, leaving weapons and uniforms behind, trying to just get home and out of the way of the KPA.

The timeline goes like this:

June 25 - the KPA crosses the 38th heading south.
June 28 - within 3 days, Seoul has fallen. The North is delayed in Seoul by the demolition of bridges crossing the Han River. While not being a deep river, it is wide and it's tidal flow is one of the greatest in the world.
July 2 - Suwon falls. The provincial capital 30 miles south of Seoul, MacArthur had visited there only 3 days prior. This was also the beginning of American contact. Task Force Smith was sent up to stage a delaying action against the North Koreans.
July 5 - Task Force Smith makes contact with the North Koreans south of Suwon at Osan. The KPA T-34 tanks are superior to everything that TFS has. 105 mm mortars, recoilless rifles, 2.34 inch bazookas, nothing can stop the KPA armored column. Task Force Smith is destroyed. Wounded and captured are later recovered, many with hands bound, executed.
July 14 - The battle of Taejon (Daejeon) begins. The city falls by the 21st, division commander Major General William Dean is captured and held in Pyongyang until the end of the war. He is awarded the Medal of Honor for his actions during the battle.
July 20/21 - Daejeon falls. 922 soldiers dead, 228 wounded, 2,400 missing. However, the goal of Daejeon was to delay the KPA while the Pusan Perimeter was established. The 24th infantry division retreats to Daegu, inside the perimeter.
August 4 - September 18 - Battle of the Pusan Perimeter. The final push of nearly 100,000 KPA soldiers is stopped by a UN force of 140,000 soldiers. Despite two massive pushes, the KPA supply lines are stretched too thin, American/UN supplies and weapons arrive, and the Inchon landing breaks the rear echelons of the KPA. They are forced to retreat.

Within 6 weeks, the KPA had taken over all of the Korean peninsula outside of the Pusan Perimeter. The fear that UN soldiers would literally be driven back into the sea was a legitimate one. The US was woefully unprepared for anything that the North could have thrown at them.

At this point, China has had nothing to do with the war other than sending ethnic Koreans home to fight and gearing up for their eventual entry.

The second part of the war was the complete opposite of the first. The KPA was retreating so quickly that the UN forces could barely keep up.

September 15-19 - Landing and Battle of Inchon
September 22-25 - Second Battle of Seoul
September 27 - US Eighth Army (Pusan) makes contact with X Corps (Inchon)
October 9 - UN Forces cross the 38th parallel
October 19 - Pyongyang falls

Crossing the 38th parallel was the point of no return. The Chinese had said that if US/UN forces entered North Korea, they would enter the war. MacArthur ignored these warnings. Six days after Pyongyang fell, ROK troops were attacked and destroyed by PLA forces north of Pyongyang. UN forces advanced north and made contact by the beginning of November and they were then in full retreat by the end of November, losing Pyongyang and then Seoul (for a second time) on January 7th.

What made the Chinese force superior were the numbers, supplies, and tactics. US forces were about 325,000 at peak while PLA forces were around 1.35 million at peak. UN forces in North Korea overstretched supply lines and were outfitted with summer gear. Manchuria, in winter, in summer gear = terrible idea. Finally, US forces were used to driving around in jeeps, tanks, and trucks - they were stuck to the roads. The Chinese walked. In a country that is pure mountains, the Chinese were able to set up ambushes along roads, hide in the woods, and travel across terrain that US forces wouldn't even think of crossing. The Chinese were able to use this and push the UN forces south again. However, once the UN forces were properly supplied, the soldiers were in-country, the UN forces were able to stabilize the front around the 38th parallel and negotiations for peace began.

TLDR; the Chinese had nothing to do with pushing US forces to Pusan, that was all KPA. Once the Chinese entered the war, they used superior numbers, superior tactics, and better supplies to push UN forces back. Once US/UN forces were able to get a steady supply line and a decent sized force, their technological superiority gave them the edge they needed to hold the line at the 38th.

For more information, I highly recommend The Coldest Winter by David Halberstam and Scorched Earth, Black Snow: Britain and Australia in the Korean War, 1950 by Andrew Salmon.