There are a few distinct questions here, so I'll answer each briefly.
What are some real world problems with real scenarios blockchain interoperability solve?
The problems solved by blockchain interoperability really depend on what problems are being solved by blockchains at any given point in time. A simple example is using an NFT on chain A to get some privilege on chain B. Suppose chain A supports an NFT marketplace, and chain B has a mechanism to manage ticket sales for a specific venue. A musician may want to sell an NFT collection on chain A, and say that anyone owning that NFT on chain A is given priority to buy a ticket for their upcoming concert, where tickets are sold on chain B. This is not possible without blockchain interoperability.
There are a ton of other examples, and this is just one.
Can't I simply create a bot and take the data and create a transaction and send it to another chain?
Suppose you have some tokens worth 5 USD on chain A, and you want to send them to chain B. You have some value on chain A, and are trying to transfer it to another blockchain. You can't just submit some transaction to chain B sending things from chain A, because chain B doesn't know anything about chain A. It won't trust you "just because".
A current solution for this usecase is to use bridges, however those are not always the most reliable, and historically have been very prone to being hacked. Additionally, there is substantial lag, because chain B's side of the bridge needs to make sure the transaction on side A is finalized before giving you the assets.
What does the relay chain do?
The fundamental reason the relay chain exists is to share security among other blockchains, and to facilitate interoperability through parachains advancing in sync.
The shared security is because the relay chain does sharded execution of all parachains attached to it. Basically, what this guarantees is that the parachain block was executed according to the parachain's rules, which is guaranteed by the relay chain validators. So if the parachain's rules say that a transaction must be signed by the sender, and the sender must have enough balance, the relay chain will check that for every parachain block. This is more efficient than a normal blockchain because the execution is sharded amongst validators, as compared to the full replication of general blockchains. So it becomes equally expensive/difficult to attack any parachain as it would cost to attack the relay chain itself.
By comparison, if there are two standalone chains, it will cost a different amount to attack each, based on the amount staked on each chain (or the collective mining power, depending on its consensus). This means that parachains are generally more secure than standalone chains.
The interoperability comes through XCM, which is the language used for these parachains to communicate, and XCMP, a protocol executed by the relay chain to pass trusted messages between parachains. The communication can be faster than between sovereign chains, because parachain A can act on a transaction in parachain B immediately after it occurs, knowing that if one gets rolled back, both will get rolled back.