So, now that I've been on this game a little longer, I've been seeing the word straws more often, and I've been trying to piece it together but I'm still slightly confused. What are they and what's their purpose? Is it frozen semen from stallions? (cause that's what my first guess was)

You can find this answered in the game help section:

Semen Collection
The Horse Eden Veterinary Office offers a semen collection service. Collectyour favorite stallion's semen in straws for later use. Or to sell to other members. It costs 2500 to use a straw. You may collect up to 30 straws from each stallion each year. A stallion may not have more than 50 straws in existence, whether you own the straws or other members own them. Foals produced from straws do not count against a stallion's yearly foal count.

You may sell your straws several ways. You can place them for sale in the Store. Keep in mind that any Premium member may purchase them and use them on any mare. Straws not sold within 7 days will be take down from sale by the game. If you want to sell straws selectively, then you can mark the "Straws Available" box on your Stallions page under "Put Horse Up For Breeding". Another option is to sell straws in the Forum, where you can create an advertisement.

Thank you! Makes sense now.