It'd be great to hear more about what your company thinks the "devops way" is.
If we look back in history, the first software engineers actually were operators as well (the women who programmed the ENIAC without any kind of manuals because hardware engineers were basically 'We built it, you figure it out'). Over time as computers became ubiquitous and affordable to everyone, software engineering broke out into specializations like system administration, system operators, and developers. Where once the hardware engineers were pushing the responsibilities on to software engineers, now the developers were doing the same to the other specializations. This led to some increased silos due to organizational structures and the ever increasing battles for budgets. System administration specialized into network, operating systems, security, and database administration. Development has specialized into hardware, front end, back end, quality assurance development just to name a few! As we see the specialization because each of the components become more complex we see more siloization.
So what does DevOps have to do for any of these groups? Bringing us back together to find the common grounds and the differences in our specializations that help us inform how we do our jobs for the better of the organization. We can see how some of the benefits of coding has been shared with operations folks in advancements such as infrastructure as code. With development, qa, and operations folks working together we can see continuous integration, continuous delivery, and continuous deployment become possible.
What does DevOps specifically have to do with front-end development? From what I've seen, to some degree the siloization is still happening and many front-end developers don't think they are part of devops even though devops with front-end AND back-end development at Flickr with operations was the key point that started the drive for cooperation and collaboration between groups because the magical outcome of 10 deploys was possible when these parties were on the same page! Key elements like defining your sandbox environment with abstractions that are provided with test kitchen, vagrant or cloud provisioning, and chef for example will allow a front-end developer to write deployments that can easily be deployed to OS platforms of ubuntu OR centos locally or into systems hosted in AWS or Digital Ocean without changing the code.
The impact means faster delivery of code into environments where customers can see the benefits and changes more quickly. Often this leads to a change from trying to get the code out to be seen, and more about the ability to choose when those changes get deployed because it's possible to do the deployments so quickly.
Andy Glover from NetFlix gave a keynote at the recently-completed UberConf in Denver, CO. Apparently, they have worked very hard to make deployments at NetFlix so boring that it becomes a non-event. Andy claimed that they do 4000 deployments to production PER DAY at NetFlix. By comparison, I have heard it said that Google deploys to production every 7 seconds and Amazon does it something like every 12 seconds (11.6s according to this: https://news.ycombinator.com/item?id=2971521). @4K per day or roughly once every 21.6 seconds, NetFlix still has some work to do ... (tongue in cheek, although I think with the culture that they have, they probably do think they still need to improve).
Another thing Andy said about NetFlix was that they only hired senior developers (no junior devs or interns), everyone was allowed to make mistakes because it's the best way to learn, they have a very flat org structure, teams are very independent of each other and determine their own course and destiny, and you're expected to perform above expectations at all times (Andy confirmed that at NetFlix, meeting expectations == a very generous severance package). Oh, and everyone is expected to come in to the office to work -- teams are all co-located in Los Gatos.
While most of us can only dream of working at a place like that, where technical excellence and speed are top priorities, this description of the culture at NetFlix kind of hints at some of the things you need to be successful at DevOps.
What does this have to do with front-end developers? Well, IMO, moving to DevOps affects front-end developers and all other kinds of developers in that they need to be able to respond very quickly to change and that will never happen unless they work constantly and seamlessly with the rest of the folks who make CI / CD possible and painless. DevOps takes the boundaries of "team" as you probably know it today and pushes it out to encompass not just front-end developers or even just developers. The concept of "team boundaries" gets blurred and expanded considerably with DevOps and the challenge is adapting the culture and behaviors so that "the team" remains cohesive and productive.