Why constrain the message lengths in indistinguishability in the presence of an eavesdropper? I need your help with a very basic concept in cryptography which I can't understand/prove on my own.

I'm trying to prove and understand why, under "indistinguishability in the presence
of an eavesdropper" encrpytion, it is required that the lengths of the messages output by the adversary satisfy $|m_0| = |m_1|$.

Just to make sure the reader understand exactly what's I'm talking about, the context of my question is page 6 of these lecture notes on Constructing a Computationally Secure Scheme Pseudorandomness (“Claim of indistinguishable encryptions in the presence of an eavesdropper”).
Well, the problem is that -- as long as your encryption algorithm accepts messages of arbitrary length -- it is impossible to hide the length of the encrypted message.

This is actually quite natural. Think about it, when I encrypt a 1000 bit message I simply need more bits to represent the ciphertext than when I encrypt only a single bit.

Therefore, if one of the messages is allowed to be longer than the other, it is in general quite easy to distinguish between the two cases, because in the one case the ciphertext will be longer.