Speaking at a Robert Burns-themed celebration in Edinburgh, the First Minister said she understood the "resentment" at having to apply for the right to stay in the UK.
She promised the country would continue to welcome EU nationals, while announcing more funding for the Stay in Scotland campaign, which has supported more than 4,000 EU nationals with their settled status applications since its launch in April 2018.
Under the UK Government's settlement scheme, EU nationals can be granted either settled status or pre-settled status for those with less than five years' continuous residence in the UK, or reject the application.
Ms Sturgeon acknowledged the additional £10,000 from the Scottish Government "is not a massive amount of money" but could make a "real, tangible difference".
Expressing "profound concerns" about the settled status scheme, Ms Sturgeon said the additional money would help people with their applications by enabling organisations such as Citizens Advice Scotland to "handle more referrals of complex cases".
"People look at what's happened over Windrush, they look at the hostile environment that the UK Government created, they look at some of the rhetoric around UK immigration policy and it's understandable that they feel concern about their future here," she added.
"Thank you for doing us the honour of choosing Scotland as your home. Thank you for the contribution you make to creating the country we live in today.
"And to say very clearly, unequivocally and as loudly as I possibly can: You are welcome here.
"This is your home. It is as much your home as it is my home and we want you to stay in Scotland."
Amid concern about the impact of Brexit on the size of Scotland's population, she said: "You will always be welcome here. We're honoured that you've chosen to live here, and we want you to stay."
She added: "If we look at Scotland's demographics over the next 25 years, we need to keep encouraging people to come here if we're to keep our working-age population growing and making the contribution to our tax revenues that we need."
"We've been reassured that Scotland is our home and this is exactly what we wanted to hear.
"There have been voices around since the Brexit referendum happened that we are not welcome but it never came from the Scottish Government."
Ms Ingle, who works for the charity Settled Scotland to help vulnerable Europeans keep their right to live and work in the UK, said many people were fearful of what might happen after Brexit.
"We are lucky we are in Scotland because we've been reassured over and over again that this is our home and we are welcome here," she said.
"Scotland is my home, my children go to school and nursery here, I work here, my husband works here, so we are part of society and thank goodness we feel welcome here."