I remember this. I get these twinges at the beginning of almost every poly relationship.
It sounds like you're having "mine" syndrome--you've found something so precious and wonderful that you want it to be all yours. It happens. It's natural. It just is the big roadblock to you being in a relationship with this person that you've come to care for so much.
Before I go any further, I have to tell you this: if you go in with any intention of "converting" her back to monogamy, you are most likely in for a world of unpleasantness. This woman isn't splitting her heart between you and her current partner(s), she cares for each of you fully and independently from the rest--so it's not like you can win more "portions" of her love. Doing anything to pull her away from her current partner will break her heart.
My advice is this: spend some time in deep thought, and figure out if you think there is a way you can be with this girl. Consider this: when you're with her, do you feel like she's withholding anything from you? Does dating her feel any different from dating other monogamous people? Did you feel any of these pangs of hurt before you saw her with her boyfriend?
If you answered "no" to these questions, you have to realize that she does care for you, that she is opening her heart to you. Like I said, poly people don't split their love, but they love each partner in equal share. Her relationship with you will not be diminished by her relationships with other people. Concentrate on this fact, first.
Also, perhaps spend some time befriending her other partners. Once you get to know them, you will get a sense of what she loves in them too. Developing respect for those partners can go a long way for slaying the "mine" feeling--because you understand their role in her life, that what you offer is different from what they do, and that you enrich her life. And be honest with her--let her know that you just aren't ready to see her having PDA with another person because you're so new to all this. You can't expect for her to hold her lovers at arms' length forever, but if you openly explain to her your feelings, she'll almost definitely cool down on the PDA to make you more comfortable.
That's about it: realize that what you have with her is special, get to know the other legs, and be honest with the girl. This is the foundation of poly in a nutshell.
I hope this helps, and good luck getting through this.