What is multi-prime RSA (RSA-MP)? I found some related question but no real explanation of what it is and when and why to use it. What are the benefits and downsides and is it recommended?
Multi-prime RSA is simply using more than 2 prime numbers in generating RSA public key - The public modulus would have more than 2 factors. 

We use it because it has more efficient key-generation and decryption/signing operation, which is the benefit of it. 

The downside being it might be easier to factor a multi-prime RSA public key than a dual-prime one. 

It's neither recommended or recommended against, it's simply an possible option without any endorsement.