Quite simply, they are looking for a man who is different than the local men.
These are the biggest complaints Armenian women have with dating and marriage to Armenian men: After you have read the complaints Armenian women had about local men, you can start to understand how and why many women would like to meet someone different.
It's generally much more work for an Armenian woman to get married if her virginity is not intact.
Here is a link to a thread on the Heyforums about Armenian women and virginity.
Quite simply, a lot of these women want to have a life more like a western woman's.
While they may not want one identical to a western woman's, they do seek greater kindness, love, affection, and better treatment in their relationship by marrying a foreign man, which is not unreasonable for any person to want. You get a girl who is very monogamous, so you don't have to worry about being cheated on, likewise it's important to remember that many Armenian women remain with their husbands even in the face of domestic abuse.
They are also responsible for passing on traditions from one generation to the next.