How do labs and scientists work?

Updated after discussion in the comments

Obtaining a scientist will reduce research time for projects in the Research Lab. However, there's no way in game to see this benefit other than research time goes down.

You can additionally build a lab where you can assign 1 (or 2 if you pay to upgrade the lab) to further boost the research time reduction benefit. Also to your question, building the lab doesn't provide you any additional scientists, it only allows you to better utilize the ones you have.

Presumably this research benefit also helps when conducting research in the Shadow Chamber, but I haven't been able to confirm.

You might be able to assign scientists to other buildings later in the game, but I haven't made it far enough to confirm that. My assumption is based off of the View Scientists screen that shows unassigned Scientists as AVAILABLE:

The Laboratory (you can only build one) functions in a same way as the Workshop: It makes a scientist count as two scientists when he or she is assigned in the Laboratory (you can assign two after upgrading), just like Workshop creates two drones per assigned engineer.

Having no more than two scientists but both of them being assigned in the lab will decrease research times for the same amount as having four scientists but no lab.

There's also a continent bonus that decreases research times just by having the laboratory built. It gives 20% faster research rate when it is constructed, giving it some additional boost.

In terms of how important building the Laboratory is, well I'd say with difficulties other than legend not very. The bottleneck is more often lack of supplies/alloys than available tech. With legend difficulty however research takes so much longer that you will have much more supply drops before you get the next level gear unlocked. The available tech will be a more significant bottleneck than lack of materials. Therefore having either a lot of scientists or the lab if you did not get many is a necessity. For example, magnetic weapons research with 0 scientists takes over 60 days in legend, over 40 with one scientist and still over 30 with two scientists.