They're changing laws now because they know it will likely go through local courts, state courts, federal courts, etc., and then end up being argued in front of the Supreme Court.
The Supreme Court is now stacked in such a way that the justices could very well overturn Roe v Wade (which is the case that set the precedent for allowing abortion and restricted individual states from banning it... argument was based on individual right to privacy IIRC).
Why do people want to ban it? Complicated to explain to a secular British audience.
Religion is still very much a part of public life in the United States. I grew up in Massachusetts, which a lot of people consider to be a turbo-progressive place. But I was raised by a Christian mom and agnostic-ish dad. Mom took us to church every Sunday, we went to Sunday school as young children, and as teenagers we went to weekly church youth groups to talk even more about Jesus. Many of my friends and family growing up were Catholic and had CCD during the week, church, and if it was a first communion or confirmation year, additional church stuff to prepare for that.
Go further south, to places like the Carolinas, Texas, Alabama, and you'll literally see a church on every street corner.
When prayer was banned in schools, on the grounds of separation of church and state, that was controversial, and some people are still bitter about that. The pledge of allegiance still says "one nation, Under God". The money is printed with the phrase "in God we trust". The president closes every address with "And God bless America".
It is still considered socially awkward, in a lot of parts of America (and consider the size of America vs Great Britain) to say you don't believe in God, and think the stories in the Bible are myths.
So... still lots of religion informing public life.
There are many, many people who are straight up raised to understand that abortion is murder. They're told that women who seek abortions are irresponsible and selfish. They're told that women who don't want to raise children can just give them up for adoption.
Adoption is treated more commercially in the States. Sometimes, it's through the state foster care system. But there are also charities coordinating adoptions between families (usually well to do) and birth mothers (often very young women). It really can come across as though you're buying a baby. Look at the movie Juno as a reference; it's satire, but not that far off the mark.
The trauma of childbirth is not discussed. I didn't learn about vaginal tearing until I was in my early 20s, and I read about it on an Internet forum. I didn't know a C-section wound could hurt so bad you couldn't sit up until I was an adult: C-sections were described as something you could just get if you didn't want to experience labour pains. I didn't know all the different risks being pregnant can raise until I was an adult and pregnant with my own babies.
The bond a pregnant woman can develop with a baby (or lack of bond) isn't discussed. The trauma a child can experience having been adopted isn't discussed.
There are states that still limit sex education to a discussion about abstinence and STDs. Sex in some places is still treated as a sin unless you're married, rather than a perfectly normal urge to have.
So that's some of the context of white American pro-lifers. As far as I understand them, anyway.
That context has been exploited by the Republican Party to gain power and votes since Ronald Reagan was president (moral majority).