Quantum mechanics is best understood with a solid grasp of classical wave mechanics.

If there were two water waves in different locations, it's easy to keep track of them, even if they momentarily cross. But if there were two travelling together in the same direction -- is that still two waves? If they then separate, which one is which? This is what indistinguishability means.

Photons are fundamentally just bumps in the global electromagnetic field. When the bumps are well-separated, we can say this is bump 'A' and this is bump 'B'. When they are close, or moving together, or interfering ... those labels are not possible.