For me, it started when I walked into the local Radio Shack in my hometown. I originally took my scanner in to see if it could get the scanner bed glass cleaned.
From the first time the guy behind the counter saw me, which he later admitted to me, he just felt 'pulled' or 'connected' with me somehow.
I had to come in a few times after that for things I had to pick up pc-wise (otherwise I would've had to travel 30 minutes north or south to the nearest large towns that carried anything).
We became friends and spent time together, and I was just starting out with my spiritual journey. The real sort of "big bang" moment I had was when we went to a Healing & Recovery store about 45 minutes north of where I live. Stones, crystals, healing music, etc. But it was the moment I saw a small figure of Quan Yin in a glass case and asked about it that I was just overcome with joy. So much so that I started sobbing. The guy I'd asked about the statue talked about when he'd been travelling and seen a lovely statue of her somewhere, but I was barely able to keep myself held together long enough to let him finish telling his story. lol
My friend sat down beside me and I just cried into him. The moment passed but I was just astounded to have had such a moment of 'this is right'.
On the way back home, my friend admitted to me that, when he got up that morning, he had a vision. He knew that I'd be crying at that store, but that, regardless of it and he not knowing why, we did have to go.
Suffice it to say, despite such innocent beginnings, we both discovered together about a past life we'd shared together.
Now, anyone can tell you 'oh you could just be making this stuff up, he could be lying about whatever', but, here's the thing. When two people who haven't known one previusly (in the current life), and who both remember a past life time and place, as well as specific moments and details of then, that there is no way to fake this. And when a perfect stranger crosses your path and is able to remember the same things you do, you just know.
Just like if you meet someone you didn't know, and you both remember a specific day at a kid's park that happened years back. No way to prove it happened, but you both remember the same things and can finish one another's sentences about what's happened.
Of course, this has only happened to me a handfull of times. The rest of the time, memories from my past lives just sort of, for lack of a better word, bubble up into my conscious mind. For me it just starts as a feeling of 'I've forgotten something. I KNOW I have. It's right on the tip of my mind... I can't grab it but I can feel it's there'. When I stop trying to reach for it and grab it, the memories come to my mind.
They're like remembering things from when I was little in this life. I may not always be able to place the exact when and where they happened, but I know in my heart that that's my past, that that was me, that those things happened. For good or bad that's remembered, I'm glad I've been able to remember so many things.

Goodness and look at me rambling on! I guess to try and answer this question in the shortest way possible, how does one know they have had a past life?
You remember it. Just like, how do you remember what happened at your birthday party when you were 7? You remember it. You don't try hard and force yourself to remember it, you just sit down, relax, meditate on it and it'll come to your mind.
Are there certain indicators that point to a past life?
For me, it's always been more like a feeling, aside from the recalled memories themselves. Something is just familiar. Like, for me, ever since I saw a picture of a Japanese building back in jr high school, something deep in my gut, I could feel it. It just felt 'right'. I felt connected and drawn towards it. I couldn't explain why and didn't understand it until I was on my spiritual path. Long version short, I'd lived a LOT of my past lives in Asia or Far east-esque lives. That's why I have always had such a deep connection with Japan and China. It's not something you think about, it's just a feeling. A gut feeling.
How did I know I had a past life?
The memories came to me. That's how I know. And in the few times I've crossed paths with someone who'd been in a past life with me, that just confirms it beyond any doubt. It's an awesome feeling when you get to talk with someone else and you're both filling in one another's sentences. Not guessing but knowing. You're both remembering the same things.

At least, that's how it's been for me.

Peace and love to everyone here <3