It's about unenumerated rights and interpretation of liberty under the 14th Amendment.
> nor shall any State deprive any person of life, liberty, or property, without due process of law
So in the past the court said liberty includes a right to privacy, which also includes things like having a right to buy and use contraceptives (Griswold v. Connecticut). This was extended to include women having a right to an abortion, with some qualifiers (the right was not unlimited, it said states did have some interest in protecting both the mother's health and fetus health).
The current court decision says that, while those other unenumerated rights have been found, they're different because they don't involve an "unborn human being".[1]
Thus they imply that a few cells (under some state laws this would be from the moment of fertilization) have rights that supersede (or at least conflict) with an actual person's right to have their liberty protected from the State.
The Louisiana state legislature has a bill introduced right now that seems to make abortion homicide, both for the mother who receives the abortion and anyone who administers it. [2][3]
[1] https://s3.documentcloud.org/documents/21835435/scotus-initi...
[2] https://www.legis.la.gov/legis/ViewDocument.aspx?d=1276214
[3] https://www.theadvocate.com/baton_rouge/news/politics/legisl...