I don't really see that it matters at all what part of speech "stretch out" is - the average English speaker wouldn't know anyway. What matters is what it MEANS.

If a person stretches out their body they are making it longer - not all curled up and trying to fit into a small space.
If one has a rubber band, one stretches it out to fit over something so it snaps back to hold the thing together.
If a garment shrinks in the wash, we try to stretch out that garment during the drying process to make it bigger again.
Similarly, if we are planning on having two sausages and a potato for lunch and an unexpected guest arrives, we stretch out the meal i.e. add more vegetables, or some beans or another sausage, to make it bigger - big enough for two.
If we stretch out a sentence we are trying to make it longer.

e.g. The cat sat on the mat. Brief simple sentence. We could add an adjective: The black cat sat on the mat. We might add another: The black cat sat on the woolen mat.......etc. etc. we are stretching the simple sentence out by adding detail through the two adjectives.