Bokura wa Kaseki Horidaa (English name: We are Fossil Holders) doesn't launch in Japan until next week, but Siliconera got its hands on a copy early, and wrote about the experience of excavating one of the in-game fossils; apparently, it's a little more complex than casually chatting to an owl in a bow-tie.
There's a range of tools and techniques required to successfully excavate fossils without damaging them, including an X-ray device (to detect what lies underground), a hammer (to break down rocks and soil), and a drill (to chisel away smaller pieces of earth). Blowing dust away (using the DS's microphone, a la Spectrobes) is also an important part of recovering pieces for your dinosaur army.
Beware, however, for if greenish cracks appear on the fossil, that means that it's getting damaged, and could break. We'd recommend reading Siliconera's write-up, as it actually sounds like quite a tense (and delicate) process!