Why larger input sizes imply harder instances?
Is it the case that time complexity is always an increasing function
  in the input size? If so, why is it the case?


No.  Consider a Turing machine that halts after $n$ steps when the input size $n$ is even, and halts after $n^2$ steps when $n$ is odd.

If you mean the complexity of a problem, the answer is still no.  The complexity of primality testing is much smaller for even numbers than for odd numbers.