@smarx How does websites like bittrex/ coinbase enables 'sendTransaction' methods ?

They don't do anything like that, to my knowledge.

Then how am I able to send Ether from my wallet to somebody else's wallet?

@smarx I mean, when I transfer 1 eth from account 0x123 to 0x456 (eg.,) how bittrex is able to interact with ethereum network to complete this transfer? My understanding is - function transferFrom(address from, address to, uint value) returns (bool ok); - function is called which is exactly similar to my 'submitClaim' function in the contract. I'm not able to call the function 'submitClaim' from html application.

I have no idea what you're trying to ask. They just send a transaction, probably via their own node. They certainly don't give their private keys to someone else.