The Swans have been rocked by sexual assault allegations that emerged early last week.
McVeigh was an assistant coach at the second-placed Swans, who start their finals campaign next week, until he was appointed Essendon boss on Monday evening.
As part of an understanding with Sydney coach Dean Cox and football boss Leon Cameron, McVeigh finished up immediately to start his new role - a common situation for coaches appointed before the end of the season.
"The dialogue with Leon Cameron and Dean Cox was pretty clear throughout this process," McVeigh said.
"If I was to win it, we know the enormity of the job.
"We sat down as it got a little bit closer to the decision coming, and if we were able to win the role, we knew the Sydney Swans, if they were good enough to get to a grand final, would be (still playing for) six or seven weeks.
"So a long time, and that was just too long if I was to win the role.
"So they were fantastic in supporting me, Leon Cameron and Dean Cox.
"Although it's tough leaving the players and the club in a really difficult moment, the job is just too big and I have to start straight away."
The new Essendon coach doesn't expect to be spoken to by police in regards to the events at East Melbourne's Pullman Hotel in the early hours of August 17.
Five players - Isaac Heeney, Chad Warner, Nick Blakey, Riley Bice and James Jordon - were last Wednesday suspended for the rest of the season for "breaching club standards".
The sanctions are separate to the police investigation into sexual assault allegations.
No charges have been laid and AAP is not suggesting any wrongdoing on the part of the five suspended players.
"It was a heavy feeling for the people not involved, no doubt," McVeigh said.
"And I think we've got to respect what's going to take place in the future.
"But (for) the people that weren't directly involved, it was a heavy thing to take, no doubt."
Asked of his reaction at the time news started to emerge that morning in East Melbourne, McVeigh said: "Just shocked, shocked, absolutely shocked.
"And I think Matthew Pavlich, the CEO of the Sydney Swans, has spoken to that. Really disappointed and shocked."
McVeigh said he hadn't felt compelled to call Essendon president Andrew Welsh or chief executive Tim Roberts when news of the Swans saga broke to make it clear he wasn't involved.
"I didn't feel compelled. I felt comfortable," he said.
"And I think the Swans got on the front foot and delivered the news to the appropriate people about 'look they're not directly involved at all'.
"So I felt comfortable. I didn't feel moved to ring the footy club at all, I was okay.
"I'm not here to speak to the Sydney Swans culture. I'm here for the Essendon Footy Club.
"They'll address that. They (Sydney) know who they are and what they're about.
"But I'm here to deliver an outcome for our club, the Essendon Footy Club, and I'm here to align us as one."
1800 RESPECT (1800 737 732)
National Sexual Abuse and Redress Support Service 1800 211 028
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