In order for a project to take effect, you must have a positive manufacturing output, you must have some fraction of that directed towards social manufacturing, and you must not have anything else in the planet's build queue. I.e. in the 'Govern Planet' screen, you must set the manufacturing fraction to some number greater than 0% and you must set the military production slider to something less than 100% military.

You must also have researched the proper technology to enable the project, but I don't think that that is related to what you're asking since you seem to be asking about the effect rather than obtaining the option. If you are asking about obtaining the option, then the Research Project is enabled by researching Institutional Research, Persuasive Research, or Research Array as appropriate for your tech tree while Birthing Subsidies becomes available upon researching Xeno Biology. The Economic Stimulus and Cultural Festival projects should be available from game start. If Stardock ever decides to reenable the Culling, Summit, and Sanctuary Projects, those would be unlocked by ideology traits in the Malevolent, Pragmatic, and Benevolent trees respectively (based upon in-game text, Culling would be from Unforgiving in the Motivation line; Summit and Sanctuary would likely be at the end of the Negotiator and Affinity lines respectively, based upon the prerequisite in the XML file).

Note that Xeno Biology is not found within the Yor tech tree and so any empire using that tech tree will not be able to use the Birthing Subsidies project, though the empires will be able to use the Assembly build options to generate population whether or not they are synthetic (or at least, they could in v1.03; I'm not sure if this has changed in 1.1 or not as I haven't checked). Also be advised that as currently implemented the Research Project and Economic Stimulus are both traps if you dedicate more than a very small amount of production to manufacturing; while debatably worthwhile to avoid the headache of going through each planet and reallocating production to allow the improvements to be upgraded, using these projects and dedicating anything more than a very small amount of production towards the project will reduce the output of the type corresponding to the project rather than increasing it. The Cultural Festival and Birthing Subsidies do not have this issue as influence and population growth are not based off of production whereas manufacturing output, research output, and income all are.

As well, unlike "free growth" provided with tech modifiers or building, the output of birthing subsidies is directly related to your planetary mnufacturing output.  therefore on manufacturing focused worlds, there is certainly a time where using birth subsidies for several turns to massively boost population, is well worth the sacrifice of not building/upgrading buidings.  you can only build/upgrade 1 building per turn, but you can generate over 10 pop per turn with massively large amounts of manufacturing (which of course becomes even more massive, and you can either redirect it back into even more birthing subsidies, or roll it over into military manufactuing).

Use of birth subsidies right away?  Definitely no.  But I would recommend once half your planet is upgraded to factories or your max tech, including a power plant and a good hub layout (for a manufacturing only world), that then using birth subsidies becomes quite worth it, especially for high class worlds which would be able to support an extremely high pop.  Switch between birth subsidies, and building/upgrading a farm to stay under the pop limit.  Can even throw in an approval building to keep the peeps happy.  Otherwise, it could take atleast a few hundred turns for that world to naturally reach a massive pop.

Can you do this with all planets?  No, not in the beginning.  You definitely need steady balance of social/military/population growth in the early game.  But mid game, especially for conquered worlds, this is how I like to play it.  Focus solely on developing the planet for a few dozen turns, and put it "online" to the military industrial complex, when it has reached very very high production levels.