Knights Templar can refer to
a) a group of crusaders from the Middle Ages
b) or a group in the south of France
c) or a group of associates of 'b' who consider themselves allies or part of that group.
Following the killings committed by Anders Breivik, who mentioned the group, some people claimed that group does not exist.
Breivik and Tarrant both give enough information that, although their words may not be entirely accurate, it is clear which group they refer to and want to ally with.
Many people with no connection to white nationalism or fringe politics know the group Breivik was referring to, and respect its members. They are famous as soldiers still today, and that is part of the appeal they have to aspiring young 'warriors'.
This page focuses mainly on 'b', the group in the south of France. The links above are not selected with any political motive, and are not perfect, but they give some background / context. Each of the links contains material that some people would say is not true.
The descendants of Templars who live in France are respected by most groups or individuals who deal with them. They are known for producing outstanding soldiers, spiritual leaders, etc. They generally have a diverse metaphysical education that encompasses and understands the traditions of most western systems, including a lot of esoteric traditions.
They are very scrupulous in philosophical issues, which appears to be what has led to their association with white nationalists. In other words, if a young person goes to them with a cultural sickness, he or she will generally leave with a cultural cure. That is not to say that 'shooting' is the cure, but rather the shooters who have interacted with them are not acting out of more ignorance than they had before meeting them.
As a general point, the group that these shooters are dealing with has quite a bit of history with many aspects of the melting pot.
Their opinions and whatever they share with visitors is based on history that they are up to speed on. It's not so much based on another group's melting pot 'political correctness'. The important point being that what the young shooters learn from that group has more value than what they learn from comparable people in their own melting pot societies. The problem that leads to shooting is not rooted in the facts from that 'Templar' group but the fictions from the shooters' own melting pot.
A lot of articles have been written on these groups, with a lot of information. Here are just some links from the internet.
An interesting killer who acted with a similar motive was the assassin of Mahatma Gandhi.
If a person is looking for a reason to shoot people they can always find a pretext.
At some point people will have honest risks to take, ways to follow legends, when people start colonizing outer space. Until then, the wars and authorities that young people in most western societies follow are fraudulent.