How can I get a reference letter if I was never "close" to any professors?

Unfortunately you've left yourself in a tight spot. This is actually one of the biggest pieces of advice I give to undergrads: if you're thinking of grad school, build relationships with professors. Even to the point of skipping a class you would prefer more in order to take another (still-relevant) class with a professor you're trying to build a relationship with.

Now that you're here however, I don't think you're in an impossible-to-salvage situation:

• First off, see if any of your letters of recommendation can come from non-academic sources. In my experience it's not uncommon for schools to want, say, two of three to be professors and would take a third one from your current boss (assuming your job is in any way relevant to what you want to study). That would get you a third of the way there.
• Secondly, begin with any professors you took more than one class with - that sort of thing tends to stand out unless they're huge sections with too many students to notice them all. Seeing your name on papers and grades twice may have cemented you in their mind enough that a gentle reminder will get you there.
• Third, turn to any professors you did particularly well with, or had very small classes with. Did you have a project and presentation that might have made you stand out?
• Fourth, look for professors whose classes are particularly relevant to your desired area of study. It may not help you much with being recognized, but the fact that your program fits their interests so well might inspire them to dig a little deeper.
• And finally, if at all possible, go visit your professors in person to ask. They may be more likely to remember you if they have both your face and name to go with your request. If you have to do it over email, include some details to help them out; tell them exactly what class you took with them and when, maybe include a link to your Linkedin page or webpage so they can go see your picture if they want, etc.

However you end up doing it, simply explain that you weren't originally planning on grad school and have been in industry for a year, but now you really want to further your studies. Most professors will be inclined to help you out as much as they can, or at worst they should be willing to tell you if they don't think they can write you a good letter, giving you a chance to ask someone else.