This question involves the human mind's memory, so to start, let's play a quick game. Look at the line below for 1 second, and then look away immediately.
G R T A O H V I L P
Now try to recall as many of the letters as you can.
Now do the same thing for the next line (look for 1 second):
Fish Table Plate Water Mouse Cookie Scroll Piece Fan Ball
Recall as many of the words as you can.
The number of letters you were able to remember is probably close to the number of words you remembered. This is because human minds chunk many small things into bigger things, if given the opportunity. Words are just letters being chunked together. Note though, that if you didn't know what "Cookie" is, chances are you wouldn't have remembered it (since your mind wouldn't have had the opportunity to chunk C,O,O,K,I,E into one word).
This process can be applied to chess positions. You want to give your mind more opportunities to chunk pieces ("letters") into positions you can memorize ("words"). The key to doing this is pattern recognition; training your mind to immediately recognize positions (or immediately chunk certain arrangements of pieces together). Pattern recognition is just like learning new words in a language.
The best way to improve pattern recognition abilities is to look at many chess positions, over and over and over. Solving puzzles is a great way to do this since your mind is actively thinking about positions, allowing them to be stored and accessed easier from long-term memory. This not only helps your chess playing abilities, but also how well you can hold many positions in your head. If you can immediately see arrangements of pieces and know that they're a certain position (like C,O,O,K,I,E means "cookie"), then all you're really doing is holding big words in memory.