A: Sneak+pickblock! (Shift+middle-click by default.) To then place them, just right-click somewhere.
Q: Why do we need two block carrying mods?
A: I released my block carrying mod outside of Curse in January 2017. The developers of the other popular carrying mod, Carry On, created theirs in July 2017 - and were not aware of the existence of mine. Thus, two mods now co-exist.
Q: What are the key differences between Carry On and Charset Block Carrying?
A: The former allows you to carry entities - Charset only allows carrying of blocks for the time being. Charset's version of the tweak also requires explicitly marking blocks as compatible and provides additional compatibility/tweaks mentioned above.
Mod developers, please refer to this document for information on adding compatibility with this module. As for modpack developers, this document (module/lib.cfg part) explains how to whitelist blocks for carrying yourself.