It´s really quite easy! First off you need a landing page with an optin form for your newsletter. I recommend using vyper.io as you can easily start a wildfire by offering a referral competition.
Then, as it is a curated newsletter, send an email to every blog author of the original post and ask for a social share. Not everyone will do, but make sure to thank those who do. Great for building relationships and keep promoting each other.
Next create a fanpage on Facebook and share your landing page there. Then promote this exact post from your fanpage (about your landing page) in the best matching Facebook groups with the most users. This way you will not create many standalone posts, but promote the same post every time. Your engagement rate will rise and Facebook will rank the post higher, thus reaching even more people.
Next share the link to your landing page in matching LinkedIn groups, Reddit, Twitter, and whatever forum or website is important to your niche to spark some more interest.
If you start blogging on your own instead of just curating content, then you can automatically triple your growth rate by using quuupromote.co to promote your blog posts on social accounts from many people that are talking about that topic to their audience.
Repeat, Repeat, Repeat... you will see traction and you did not spend a dime!