I believe License production is just a way to build something another nation has researched for yourself.
it costs civs dependant how advanced that piece of equipment is compared to your own version, and after they accept your request to produce that product, you then need to select it from your production tab and make it yourself
example: I request license production from Germany, they accept.
I then select “negotiate licenses” in the diplomacy tab for Germany and select their latest tank. This will cost me 11 civs As I have not done any tank research.
They accept.
I then go to my own production tab, select equipment and select the German variant of tank, I then add the number of factories I want to put into producing it (like any home made equipment)
license production really is a tool to allow nations to diversify their research and speed up their research; if germany is focussing on armor then Italy won’t have to waste a research slot on armor as well, they simply buy a German model and research something else.
alternatively, if a nation has a focus buff to research certain equipment, another nation can buy a production license to get their own research buff to research that same equipment.
example, Germany has 1941 heavy armor, I have a lesser version researched and am in the process of getting 1941 heavy armor, but will take me 200 days to complete. I buy a license production for heavy armor from Germany and get a research buff for 1941 armor, allowing me to speed up my tank research, so now it will only take me 150 days to complete.
it can also be a meme way to spy on your enemy nations strategy and research direction.
example, I am Soviet Union, I want to see if Germany is focussing on medium or heavy armor, so I get a license agreement from them. I now go to modify licenses and trawl through all the kit Germany currently has to see what’s getting researched and what isn’t.
I hope this helps.