Darren Osborne, 48, allegedly hit Makram Ali, 51, and nine other people near two mosques in north London shortly after 12.15am on June 19 past year.
Opening the trial, prosecutor Jonathan Rees QC said Mr Osborne had driven a Luton box van from Cardiff to London the day before the alleged attack.
Another man in the group remembered the defendant - who was "constantly smiling" - saying: "I've done my job, you can kill me now", he said.
Darren Osborne, 48, killed one man, 51-year-old Makram Ali, and injured others in Finsbury Park, north London, it was said.
Osbourne also wrote a note - discovered in the van used in the assault - discussing his motivation for the attack.
"The prosecutor said the "extreme violence" was an act of terrorism created to influence government and intimidate the Muslim community".
Regarding the murder Ali, Rees stated: "The evidence shows that, although unwell, Mr. Ali was alive just before the incident and his death resulted from the defendant driving the heavy van over him as he lay on the roadway". He had meant to attack the al-Quds march but had then made a decision to attack a mosque in Finsbury Park.More news: USA Gymnastics Executives Resign Amid Sexual Abuse Scandal
He added: 'To seek to kill someone merely due to their religion is a bad thing.
Mr Rees told the court the alleged attack was "particularly horrific" because the group had gathered in the street to help Mr Ali, who had collapsed minutes earlier.
A MAN accused of carrying out the Finsbury Park attack "decided to take matters into his own hands" after growing angry at a rise in terrorism and the Rotherham child exploitation scandal, a court has heard. He accused them all of being rapists and belonging to paedophile gangs.
Osborne also complained about child abuse "being carried out by packs of Muslim men", immigration and the introduction of Islamic ideology and Sharia law to the United Kingdom, the court heard. "It appeared to her that he was becoming brainwashed".
Mr Rees said: "The contents of the note reveal that the defendant intended for it to be read after the attack had been carried out".
Mr Rees continued: "Ms Andrews noticed that the defendant became obsessed with the subject matter of the drama".
While he had not displayed racist tendencies as far as she knew before watching the programme in May, she believed the London Bridge and Manchester Arena terror attacks which followed soon after "seemed to fuel the rage inside him", Mr Rees added.More news: Montana Governor Signs Net Neutrality Order
Smartphones and computers showed Osborne viewing material from Britain First, that "campaigns primarily against multi-culturalism and what it sees as the Islamisisation of the UK", Rees told the jury.
Unemployed for the last decade, the defendant felt worthless and that he did not fit in, according to his partner.
Rees said Osborne's former partner Sarah Andrews described the father of four as a loner and a functioning alcoholic with an unpredictable temperament who took medication for depression.
Mr Rees said Ms Andrews felt their relationship had broken down during the past year to the extent that they did not really speak with each other any more. It had been found in the van and bore his fingerprints.
It also said "Islam's ideology does not belong here", and taunted the London mayor, Sadiq Khan, the Labour leader, Jeremy Corbyn, and the singer Lily Allen, jurors heard.
Speaking with "passion and anger", he is said to have told the soldier: 'I'm going to kill all the Muslims, Muslims are all terrorists.
Two days before the attack, Osborne told a serving soldier in a pub in Cardiff: "I'm going to kill all the Muslims", the jurors were told. Osborne appeared in the dock on Monday wearing a dark blue shirt and grey pants and confirmed his identity. Ali died from his injuries and others were wounded.More news: Lara Croft embarks on an adventure in new Tomb Raider TV spot
Mr Osborne from Cardiff, denies charges of murder and attempted murder.