Brown had blood on his shirt and right arm when he ran away

After the incident, Dorian Johnson told a false story in which he remained by the police vehicle until the first shot. Johnson claimed falsely that he saw Michael Brown being shot in the upper-right chest and saw blood flowing down Brown's right arm. 

Since Johnson ran away so soon, however, he did not see Brown being shot by the police vehicle. Johnson did see, however, Brown running away from the police vehicle and then saw  that Brown's shirt and arm were splattered with blood. Subsequently, Johnson used that detail about Brown's bloody shirt and arm to enrich his own false story.
  
Already in a video recording on August 9, Johnson said:

The officer then reached out and he grabbed his [Brown's] arm to pull him into the car. So now the officer was pulling him inside the car. He [Brown] is trying to pull away.  

At no time the officer said that he was going to do anything, until he pulled out his weapon. His weapon was drawn, and he said “Ill shoot you” or “I’m going to shoot”. And in the same moment the first shot went off.  

And we [Johnson and Wilson] looked at him [Brown]. He was shot, and there was blood coming from him. And we took off running. 

As we took off running, I ducked and hid for my life because I was because I was feared for my life. I hid by the first car that I saw.

Three days later, on August 12, Johnson told CNN's Wolf Blitzer that he saw Wilson shoot Brown in the upper chest. 

The officer now is struggling with, trying to hold a grip on, my friend Big Mike as he [Brown] is trying to pull away. In a minute, I heard, “I’ll Shoot. I’m about to shoot.” 

I was standing so close to Big Mike, and ... I look in his [Wilson's police car] window, and I see that he [Wilson] had his gun pointed at both of us. And when he fired his weapon, I moved seconds before he pulled the trigger. I saw the fire come out the barrel, and I instantly knew that it was a gun. I looked at my friend Big Mike, and I saw that he was  struck in the chest or upper region, because I saw blood splattered down his side, his right area.

And at that time we both took off running.

On that same August 12, Johnson also told Al Sharpton that he saw blood coming down Brown's right arm from the imaginary wound in Brown's upper chest. 

After the first shot went off, I stepped back and I looked at my friend [Brown], and I see the blood coming down his right arm, so I know that he was hit.

The forensic investigation concluded that the first gunshot wounded Brown's right thumb while he was trying to grab Darren Wilson's pistol. Some of Brown's blood from that thumb wound surely splattered onto Brown's shirt and arm. Johnson saw that blood while Brown was running away from the police vehicle and thought mistakenly that Brown had been shot in his upper-right chest.  

Another possibility is that the splatters were seen not by Johnson, but rather by the Monte Carlo's occupants, who afterwards described the splatters to Johnson.