There are a number of reasons why people oppose Barret, among them:
her stance on political issues: There is a fear that the bodily autonomy of women is under attack (see her stance on Roe v Wade), as well as the human rights of LGBT+ people (see her stance on the rights of trans people and marriage equality), and that access to health care is in danger (see her stance on the ACA). Trump and the GOP specifically choose justices for those reasons:
The GOP platform said Trump’s Supreme Court appointments would “enable courts to begin to reverse the long line of activist decisions including Roe, Obergefell, and the Obamacare cases.”
some have the impression that this is essentially a "stolen" seat. Republicans have blocked appointments by Obama for the supreme court - as well as for lower courts - with the reason being that appointments shouldn't happen during an election. Lindsey Graham for example said:
I want you to use my words against me," Graham said at the time. "If there's a Republican president in 2016 and a vacancy occurs in the last year of the first term, you can say, 'Lindsey Graham said let's let the next president, whoever it might be, make that nomination.
Trump has been very clear that he may not accept the election results and that he needs to seat a justice to successfully overturn the result of the election. Given Trumps authoritarian leanings, there is a fear of an anti-democratic coup. Barett delined to commit to recuse herself from such a case.