Before I begin I'd like to clear up the narrative - China didn't start the Korean War by 'invading' Korea. North Korea launched an invasion of the South on 25/06/1950, and Chinese involvement began later that year to prevent the North from falling under complete US/RoK control.

As for why China became involved in the war, it's complicated. The country was completely unprepared for a war of the scale that Korea eventually developed into upon the initial intervention. The People's Liberation Army was demobilising huge swathes of soldiers in the wake of victory in the Chinese Civil War, and the economy was doing about as well as one could expect after such an extended period of conflict as China had experienced by 1950. Meanwhile there were preparations for a large-scale amphibious invasion of Formosa to finish off the Nationalists, who still had the capacity to launch air raids on mainland Chinese cities.

The PLA was still effectively a guerrilla army at this time, which would have to go into combat against what was perceived as the world's most modern fighting force in the USA. China had a small navy and a token air force by the time of their intervention in Korea, the latter of which was only flying propeller-driven hand-me-downs from the Soviets who were reluctant to hand out their modern jets. Furthermore this air force was more geared to intercepting obsolete Nationalist bombers along China's coast, not dogfights or close air support - Peng Dehuai (commander of the Chinese forces in Korea) regularly complained about his land forces not being provided with sufficient air cover. This was supposed to fight against the world's largest navy, an air force with extensive combat experience which could fulfil a variety of functions, and a mechanised army that was supposed to be armed with modern tanks and artillery. The Chinese were unaware of the initial weakness of US forces in Korea until they had crossed the Yalu, and were surprised at what they found.

Add into this that the PLA had severe logistical issues at this time, highlighted by pictures from the war. Look at any photo of Chinese soldiers in Korea and there's a good chance that you'll see in their hands an eclectic mix of Russian, American, British and Japanese weapons. This example, a (almost certainly staged) Chinese propaganda photo, shows the Chinese soldiers wielding a Sten (British submachine gun design) and a Japanese rifle. These various arms will have been captured from Japanese stocks during or after the Second World War, provided by the Russians, or acquired from surrendering Nationalist soldiers who had been supplied by western powers during the civil war. To quote Mao Zedong:

In view of the fact that the infantry arms of the People’s Liberation Army are mainly trophies captured from the enemy, there is a great variety in the calibres of the rifles. Such a situation creates great difficulty for the manufacture of ammunition, and in particular for the production of rifle and machine gun cartridges, especially as our factories can produce only very small quantities of these cartridges.

Other messages from Mao and Zhou Enlai to Stalin in the early months of Chinese intervention highlight their dependence on the Soviets in this regard, not only for supplies of fuel and lorries but also large numbers of small arms (140,000 rifles and 26,000 submachine guns, plus thousands from other categories) and millions of cartridges in an attempt to ease supply chains through standardisation on Soviet armaments.

This is one of two key factors. Stalin had to spend significant time courting Mao to essentially fight this war on his behalf, whilst Mao was all too aware of just how poorly China could handle a war without significant supplies from his communist ally. After much negotiating, with Mao ultimately left disappointed as to what exactly Soviet 'aid' would entail, an agreement was reached for Chinese troops to cross the Yalu River into Korea.

The second key factor, which helps to explain why Mao was willing to accept Stalin's sub-par offerings, was MacArthur's decision to move into North Korea. This was a controversial move even within the UN camp, and if left unchecked this offensive would have brought a western-aligned state straight directly to China's border in Manchuria. This, combined with the Nationalist dagger on Formosa constantly pointed at the mainland, proved to be too much for Mao to accept. In fact, such was the Chinese concern for the US on their doorstep that their planned invasion of Formosa was cancelled in order to intervene in Korea.

This was just a very brief overview of things with an emphasis on the military perspective. Going further on the military side I'd recommend reading Millett's The War for Korea, 2 vols. (Lawrence KS, 2010), Zhang's Red Wings Over the Yalu (College Station TX, 2002) and Shrader's Communist Logistics in the Korean War (Westport CT, 1995). In addition, definitely look at the messages between the communist leaders - these were translated into English in 1995 and published in the CWIHP Bulletin, and can be found here.