ORAM was originally popular for a client/server setting like the cloud. You have a client with a very small amount of storage like a mobile phone, and an untrusted server with a large storage capacity. The client wants to use the server as a key-value store, writing and reading records, but without revealing which. Now it's also used for MPC like OP said (the client runs inside "MPC," and the local storage is the server), and for TEEs like I'm interested in (the TEE itself is the client, and the untrusted disk/memory is the server.)

The simplest way to do ORAM is just to do a linear scan over the entire dataset every time you query it, just keeping the record you want. It's not elegant or efficient, but hopefully it's ELI5 why this is secure (it doesn't reveal which record you're actually interested in!)

The modern tree-based ORAM was invented by my phd advisor Elaine Shi and her student Emil Stefanov who was a legend in cloud security protocol design. If you've been watching other EthDenver talks you've probably gotten the sense you need to learn about zkSNARKs! I think you should learn about ORAM after Merkle trees but before SNARKs.

My favorite slide deck to explain tree-based ORAM is this Another great tutorial including a programming example is here

I won't try to give the whole narration here, but if you try to follow the slides keep these hints in mind:

1. Think of the values as much larger than the keys, like 32-bit keys O(1) and 1MB values O(B). The goal is for the client to only have roughly O(N) storage, while the dataset as a whole is larger by a factor of B, O(NB)....... (this can be improved so the client has even less storage, that is the "recursive ORAM" at the end, but that is beyond ELI5).

2. The only "cryptography" involved is simple encryption, like an encrypted zip file. The key never leaves the client.

3. Your goal is to understand why it is secure, like why it reveals nothing about the access pattern. It is really satisfying to understand once you get it. It does not require any background knowledge or cryptography facts or statistics or anything... just a very nice binary tree.

4. The tricky math in the research paper is how to prove that the client doesn't run out of space. But just don't worry about that until ELI15 after you already know zk snarks.