They get it during the night as a regular thing. During sleep, the body is regenerating and basically runs through its functions. Getting morning wood is just as normal as having a dream during sleep. It's just like REM sleep. 

My boyfriend says it's caused by needing to urinate, that's why it doesn't happen everyday, but im not 100% sure 

from what I understand, it's not really as soon as they wake up. it's just their body making sure that everything still works like it should, so it's sending blood everywhere. however, I'm not male or a doctor, so I'm not 100% sure