They lie between the metatarsals and have some interesting properties. The first dorsal interoseus has attachments into the peroneal sheath and some have attachments into the long plantar ligament. They have extensive fascial connections into a lot of soft tissue around them.
Why is this important? In two words, Shock Absorption. Warren Hammer describes it best. During gait, at forefoot strike, the metatarsals spread out, this in turn loads up the mechanoreceptors in the interossei muscles.
The information relayed from the mechanoreceptors in the interossei muscles as they spread out cause a reflex contraction of the quadriceps. This contraction stabilizes the lower extremity muscles. This is known as the positive supporting reaction.
This is one reason barefoot running has been such a boom lately. Why you hear such revelations about all of a sudden after going barefoot, my lowerback pain is gone. Hip pain is gone. Calf pain...ect.
Ultimately, what you can take away from this can be summed up with one recommendation. Don't wear tight shoes that don't allow your forefoot to expand. Without room for the toes to spread, the initial impulse from the interossei muscles don't occur and ultimately shock absorption is inhibited.