It slowly started gaining some traction because of:

• Veterans returning from Afghanistan and Iraq were seen as those that placed themselves in harms way to protect the US from terrorism following 9/11. Until then, veterans were mostly seen as people that weren't smart or disciplined enough to go to college and needed to figure out something to do with their lives. Kind of like how Trump sees veterans.
• Those returning from Afghanistan and Iraq began going to college, organizing, and being more public about their service. Imagine fighting the Taliban, then coming home to go to college and they delay your payments by 3 months. These combat vets were too head strong to tolerate that and would make a big deal about it.
• The overtly injured would be hard to ignore. Imagine a 25 year old missing their legs on a wheelchair at Golden Corral, yet they're the most polite one there.
• The GI Bill was reliable, so schools that were able to attract veterans saw significant increase in their revenue. Schools competed for this by creating policies and programs that were supportive of veterans.
• Family members of the severely injured became strong advocates.
• McCain ran for President in 2008, and his time as a POW was used as evidence of his patriotic values and endurance.
• Veterans publicly socialized in tight-knit groups. Think of the ~5 really loud dudes at the bar that seemed to have an extraordinary bond, peculiar sense of humor, and confidence. People wanted to emulate this.
• Veterans would wear military clothes/items to identify themselves to other vets for camaraderie and a warning to potential threats. American bros started imitating returning combat vets as a prototype of masculinity, specifically attire and ironically beards^1^.
• Veterans with considerable training and experience would go to major protests and either do a better job than police or intimidate them. Veterans at protests would be injured by police, making them a good example of hypocrisy that the news would report on. You'd see something like, "Veteran that protected Iraqis right to protest sent to hospital by Oakland police."
• Veterans became the group with the highest suicide rates in the country...by a lot.
• The Arizona VA Medical Center scandal outed that veterans were dying from lack of access to healthcare that was purposefully hidden by administration. Veterans had long been complaining about terrible healthcare services, but were brushed off or made to seem too privileged and ungrateful^2^. When this scandal broke out, it was major validation for the vets and shame for the public and government. Politicians and citizens that wanted to avoid the shame would pretend to be veteran advocates and accuse others of the problem.
• When the government and public tried correcting this for the recent vets, all the Vietnam vets that had been ignored and marginalized started coming out too.
• The baby boomers that spit on Vietnam vets and called them baby killers had to rid themselves of their shame, so many of them became super veteran supporters. As my one military buddy put it, "They were competing for American of the Year."
• The country had a reckoning with how they had treated all veterans until this period. Basically, the country felt it had to make up for it.
• People that want to ride veterans' coat tails without the risk and effort advocate for veterans to get some of the esteem.

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1: Fun fact! The beard thing was a rebellion to having to be clean shaven every single day for ~4 years straight. It was also a way of hiding that someone was in the military.

2: I've worked in the VA. It is pretty common to hear medical providers say something like, "I wish I didn't have to work for the rest of my life," when referring to veterans asking for help applying for disability.

Edit:

Also, obligatory: Thank you for your service.

Obligatory: discomfort