I think a high-level explanation is the right thing to give here:
For recognition of a producing process (in this case: speech) from a signal (in this case: audio), you need to find a mapping from signal to a set of features.
The closer (correlated) these features are to the parameters of the producing process (here: voicing utterances), the more information your feature set contains about the thing you want to recognize; the better your recognition can work.
And since linear predictors have proven very effective in reducing the amount of data needed to transport a voice audio signal (for audio compression!) whilst keeping understandable, it is intuitive that using this thus demonstratably well content-correlated (at least to the human listener) set of coefficients for speech recognition.
It's all about reducing your signal to less data whilst keeping the maximimum of info about what you want to recognize and simultaneously dropping as much info about what doesn't contribute to the phenomenon of interest!