Good point! It’s funny, though, I look at what they’re doing and don’t see much beyond what I’m doing. But I do think they probably have a lot more backlinks built up.
I don't know what your expertise or experience is, but if they're listed #1, #2, and #3, they are absolutely doing SOMETHING. I have a sort of philosophy to apply here, and it actually transcends SEO and applies to life itself, particularly with attempting to discern the existence of God. In short, nothing happens by accident. Literally nothing.
This is particularly true with Google, which is at core, despite the illusion of everything else, a computer program. It's important to have a healthy belief in this, because it is that belief that keeps you looking, and digging, when you seem to have no other reason to continue.
So, deconstructed, what you just said was "I don't know how to figure out how and why Google does what it does.", and to that I say WELCOME TO SEO.
In practical terms, I suggest you probably need some tools. Looking at your competitors backlinks would be my first step. Those backlinks are meaningful; the trick is to figure out what they mean. I'll give you an anecdotal example. I was also in a situation where I couldn't figure out why a competitor was ranking so well, so I checked the backlinks and amongst the standard-grade citations, I found a single forum post which linked to the competitor's home page. Huh. A nothing backlink on a forum which was pretty much nothing, and for some reason I wondered if that might be significant.
So I ran a search of whatever backlinks that might be aimed at that single and insignificant-looking forum post. BOOM!
It was HUGE. Literally hundreds if not thousands of spammy, crappy backlinks aimed at a nothing forum post, which in turn aimed at the competitions home page. This was years ago, and the method/technique may not work anymore, and it's certainly considered "blackhat" and risky, however the over-arching point remains. Something WAS there. I didn't know it existed until I went looking. So my advice: Look.
In another situation, I did something similar and found that a Better Business Bureau backlink carries a lot of juice to the competitor's home site. At the time, a basic membership in the BBB was about $325.00. After I knew this, I talked to an IRL Plumber and he said that the BBB membership was the best investment he ever made to his business. The phone started ringing immediately, and (he said) he made his ROI (Return On Investment) in just two days (leaving the remaining 363 days of the year pure profit). Only he didn't know why. He thought people were seeing his listing on the BBB "website" and were calling him as a result of that. I tried to explain to him about LSEO, "juice", backlinks, etc... and his eyes glazed over and he gave me that skeptical "you talk just like all the other shady SEO types that try to talk to me" look. So I let him continue to believe what he believed. Dammit Jim, he's a PLUMBER, not an SEO Specialist. Sometimes, I've learned, it's best not to try to tell the clients certain things. And if you do, you have to make them prove they really want to know, because the fact of the matter is, none of this "knowledge" is durable. Google makes a single tweak to the algorithm, and years of Google Gospel get's flushed down the commode. I've learned it's not the "answers" that are most important, it's the questions. Asking the right question is frequently more important than whatever "answer" there may be.