Why do babies keep putting their fingers in the mouth?
Sucking on fingers is mostly the baby's way of handling stimulation. The noise, being surrounding by people. Everything is new to them. Sucking on something is a way to make them feel safe.
Obviously, teething is the most reasonable answer that you will come up with.
When they're teething (most babies start teething between 4 and 7 months) the irritation in the gums might cause some pain and a lot of drooling. They feel like they are chewing on something, to relieve the pain.
It usually lasts, till a baby turns 8 months old and, in some cases, it may continue till the age of 2.

- Being bored or hunger could also be other explanations. They might think they'll get milk by sucking on fingers. They don't understand that they need to hold the bottle first.
Once they have finished exploring what they can reach (their hands), they will then get more curious, start crawling, and grabbing whatever is around them...
And it is about that time that you start fricking out ...cleaning the floor non-stop. You'll buy some plug socket protectors and safety table corners...