Learn Spanish (Spain): Nuestras gallinas ponen muchos huevos.

Yes, poner is most commonly used to mean "to put" or "to place". However, it is also the standard Spanish verb for "to lay" when talking about birds laying eggs (poner huevos).

What is the difference between gallinas and pollos?

Una gallina refers specifically to a live hen (an adult female chicken), which makes sense here since they are the ones that lay eggs. Un pollo is mostly used to talk about chicken as meat for food, or to refer to a very young chick.

Why is it nuestras? Does this mean the owners are female?

Not necessarily! In Spanish, possessive adjectives like nuestro/a must match the gender and number of the thing being owned, not the owner. Because gallinas is feminine and plural, you have to use nuestras, regardless of the gender of the people who own them.