One way is to simply give the antagonist the same goals as the protagonist, or have them come into conflict in a relatively harmless, but still high stakes way. As mentioned earlier, characters like Apollo Creed in Rocky can be antagonists, while not being evil nor villains. This is common in sports movies or anything else where most of the team’s conflict is internal, such as Remember The Titans, and you don’t need to build up the opposing teams as having any kind of nefarious goals. All the opposing forces just want to win the championship, something that isn’t evil at all. You will want to give the antagonist some character, but usually you don’t want them to be played up as evil. Give them a relatively basic motif and just fill it in. Once again, Creed was trying to further his career, sometimes the antagonist can simply be the defending champs trying to retain their title, or someone with a superiority or inferiority complex, or something to make them unique.
The more common way of making a sympathetic antagonist is to make them someone who is fighting for what they believe in, and while they have a cause you can get behind, they’re taking it too far. A good example of this would be Killmonger from Black Panther. We can get behind his cause, wanting more resources for the third world. However, the fact that he’s going about it in such a violent way, including attempting to murder Black Panther and many of his allies, makes him an antagonist. You understand his cause, but not his methods. Doctor Octavius in Spider-Man 2 in another example. He wants to finish his experiment, and you feel sorry for him because of his wife’s death, but he’s putting all of New York in danger, and tries to kill Spider-Man. You also see him being friendly and optimistic, only wanting to do good before his first experiment goes wrong, which does endear the audience to him somewhat.
The most common way of doing this is to combine these two ideas. Give your antagonist a motif that ordinary people can sympathize with, (such as Killmonger’s cause) and give them some tragedy (like Doctor Octopus’ wife dying. You can simplify things by combining them by making their tragic moments related to their cause, but this isn’t 100% needed. I would also recommend you showing them being a “good guy”, or at least a normal person before they become an antagonist in order to let the audience connect with them just a little bit more. Although you can do this in a flashback, it’s best you start with their more sympathetic or relatable moment, because first impressions count makes a big difference.