Is this the same sign as "get"?
Yes, *obtain*, *get*, *acquire*, and *receive* all share this exact same sign in ASL. Because ASL is a concept-based language rather than a word-for-word translation of English, you use this one sign for the concept of taking possession. The specific English word is implied by your context.
Can I use this sign to say "I get it" when I understand something?
No, you cannot. This sign is strictly used for physically acquiring, receiving, or taking possession of a tangible or abstract thing (like a book or a degree). If you want to say 'I get it' to mean 'I understand,' you must use the ASL sign for *understand*.
Do my hands have to be tightly closed fists?
Yes, you should end the sign with 'S' handshapes (tight fists). While fluent signers often start the sign with slightly open hands that quickly close as they pull back toward the body, ending in firm fists is essential to clearly show that you have successfully grasped the item.