You can read about it here, an article written soon after his election London mayor: The Sadiq Khan story - BBC News. So - there is an element of truth in what you said - his opponent in the race tried to get elected in a campaign that seemed Islamophobic which probably lost his opponent votes. However he was elected because he was a strong candidate, someone the ordinary Londoner could identify, grew up on a housing estate, son of a bus driver, worked as a lawyer on cases of human rights and discrimination, held many posts in government and was all round a highly qualified candidate for the job. And as a mayor he has shown in many ways that the voters chose well when they elected him and is widely admired internationally for his work as mayor of London.

So first on the campaign and how it probably lost his opponent votes:

“Whoever came up with the approach, Mr Goldsmith's campaign focused heavily on portraying Mr Khan as an associate of "extremists" - which in turn allowed Labour to attack the Conservatives for pursuing "divisive, dog-whistle" tactics.”

“Mr Khan took this to be an attempt to smear him by association because of his religion. The Conservatives insisted they were talking about his brand of left-wing politics - but Mr Goldsmith repeatedly said the Labour candidate had "given platform, oxygen and cover to extremists".”

He was a very capable candidate, trained as a lawyer, who worked on some high profile cases to do with racial discrimination and civil liberties who had won cases at the European Court of Human Rights, the House of Lords and the Court of Appeal as human rights solicitor

He grew up on a council estate in South London, and was the son of a bus driver and married to the daughter of a bus driver, so in a position to emphasize with other Londoners in similar positions.

He had been an MP since 2005, and had several positions including Labour whip, communities minister, and then was appointed to the cabinet as Transport minister. Also shadow political reform minister. He was Ed Milibands’ campaign manager, - when he won the Labour Party leadership campaign. He nominated Jeremy Corbyn as leader though he didn’t vote for him.

So he had a lot going for him, but still was a surprising choice when he came top of Labour’s list of candidates for the mayor. He won the Labour canddiate election convincingly in every round, ending ahead by 58.9% to 41.1% against the second choice, Diane Abbott.

Anyway so the electors warmed to him, with his story of rising to mayoral candidate from son of a bus driver, living on a council estate, and the nasty campaign against him probably helped as well and he won with a convincing majority of 56.8% versus 43.2% in the second round (44.2% versus 35.0%, Green 5.8% in the first round).

His actions as Major since then have helped cement his position - for instance after the recent terrorist attacks and the Grenfel fire he is there on the scene, and often on our TV involved and doing what he can to help. He is obviously hard working, and a highly motivated mayor. He was elected because he was an excellent candidate for the job and in his action since then, he has proved that the Londoners chose well when they elected him. That he is Muslim reflects the cultural diversity of London.

As for that Trump tweet, Trump tweeted

At least 7 dead and 48 wounded in terror attack and Mayor of London says there is "no reason to be alarmed!"

— Donald J. Trump (@realDonaldTrump) June 4, 2017

What Mr Khan actually said was that there is no reason to be alarmed by an increased police presence in London (for their safety). He said that the capital “will never let terrorists win”.

Quoting from him:

"There aren't words to describe the grief and anger our city will be feeling today. I'm appalled and furious that these cowardly terrorists would deliberately target innocent Londoners and bystanders enjoying their Saturday night. There can be no justification for the acts of these terrorists, and I'm quite clear that we will never let them win, nor will we allow them to cower our city or Londoners..."

"... Londoners will see an increased police presence today and over the course of the next few days. No reason to be alarmed. One of the things the police and all of us need to do is to make sure we are as safe as we possibly can be. I'm reassured that we are one of the safest global cities in the world, if not the safest global city in the world. But we always evolve and review ways to make sure we can be as safe as we possibly can. "

Trump renews feud with London mayor over terror attack - BBC News

The video itself is here London attack: 'Grief and anger' - Sadiq Khan - BBC News

More videos by him here: Sadiq Khan - BBC News

- Trump got it completely out of context and misunderstood him. Trump's insensitive remark at a time of tragedy lead to new calls here to withdraw his invite to the UK for a state visit. And BTW the police responded within eight minutes - many more would have been killed if they hadn’t shown up so quickly.

This is New York Mayor Bill de Blasio reacting to it, saying that what Sadiq Khan said is exactly what he would say in the same situation and that his actions were exemplary. New York’s Bill de Blasio rises to London mayor’s defence

The US conference of Mayors wrote a letter in support of him writing

“The United States Conference of Mayors stands today united with Mayor Sadiq Khan of London and the people of London

“We send condolences to family and loved ones of those dead and injured.

“We send praise and absolute respect for the police officers who responded with 8 minutes to kill all three of the London Bridge terrorist murderers.

“The Mayors of the USA who haven’t had a terrorist attack know full well that in today’s age of terrorist actions, domestic and international, an attack could happen in their city.

“And the Mayors of the USA whose cities have been hit with an attack of hate and murder have an even more personal feeling for London Mayor Khan. We commend him for his statement today.

“He has risen above this crisis of death and destruction, as mayors continue to do, to alleviate fear, to bring comfort to his people of London and to give support to the first responders who continue to protect, defend and provide emergency care to his people of London.

“Thank you, Mayor Khan, for your leadership during this crisis.”

As for his politics, he speaks up for the socially disadvantaged, the discriminated against of all types, the poor. He speaks for a spirit of unity and not letting the terrorists from ISIS split us from each other.

Here he is leading the vigil in London after the Manchester Love concert: London attack: Vigil held to remember victims - BBC News

With the Grenfell tower block fire, he was amongst the first on the scene, closely followed by Jeremy Corbyn and the newly elected labour MP for Kensington.

Meanwhile the local councilors were nowhere to be seen, remaining in their offices, and Theresa May only visited two days after the disaster and she spoke to the fire officers and didn’t speak to the survivors at all on that day. She returned and spoke to them a day later after she was criticized in the media for not speaking to them - and even then only in carefully controlled situations - in a hospital, in a church behind closed doors, and she invited them to 10 Downing street. There were no videos broadcast of her talking to them AFAIK. Meanwhile our Queen went and spoke to them in person too along with her grandson Prince William in a relief center helping victims of the attack, so the idea that Theresa May didn’t speak to them for security reasons doesn’t make much sense.

So he is someone who “mucks in” and is there on the scene when there is trouble. He is never shy of being there when things get difficult or facing hard questions. The contrast with the “at a distance” approach of the prime minister and councilors in this disaster was striking.

He became a voice for the Grenfell tower community, raising their issues on our TV many times. Sadiq Khan: Grenfell Tower community is frustrated and angry - BBC News

He is one of many who is standing up for the rights of the residents, that they should be rehoused within the area, pressing for support on the scene by people with high visibility jackets (a support center was set up with loads of specialists to help them but it was far from the scene with them sitting at desks waiting for them to come and hardly anyone found it and people with high vis jackets were only on the scene several days after the disaster after many comments in the media about this).

He also says the government needs to be held to account for not acting on warnings about cladding in previous fires, and is asking for transparency in the investigations, and for them to act urgently on the Grenfell fire to make sure other tower blocks are safe, without waiting for the inquiry to finish. For instance, Sadiq Khan urges government to publish list of checked tower blocks