Why is it hard to fake proof of work?
For each new block to be fixed to the end of the blockchain, a mathematical puzzle must be solved. In simple terms, the block is hashed, along with a hash of the previous block and a 'nonce', to create a new hash. The nonce can be thought of as the 'solution' to the block. It is a value such that, when all the information (previous hash, transactions and nonce) are hashed together, the resulting hash is below a certain value. (You can see that all the hashes of blocks have a long string of leading zeroes.)

The aim is for a new block to be completed every ~10 minutes, so the network agrees on a 'hash threshold'. The more computers that are competing to find the nonce, the lower the threshold becomes. How many computers are competing is gauged by whether blocks are regularly found in less than, or greater than, ten minutes on average.

So if your Bitcoin client finds a long blockchain, where each nonce is found according to a low threshold, it can surmise that a lot of work has gone in to creating that blockchain.

Even if another blockchain of the same length existed, for it to have such 'rare' hashes a lot of computer time would need to be dedicated to its construction. The longer the (real) blockchain gets, the harder it is to construct one that is equally long and has equally low hashes for each block.