"Let's" is a contraction of "Let us". Expand that, and the sentence is pretty straightforward.
- Don't let us get you cheap
The meaning here is the literal meaning, but to understand it, you have to see the line in context. The character who says this, Barbara, is trying to get the other character, Bill, to mend his ways and join the Salvation Army.
to not win cheaply
In the quote your sentence comes from, she's goading him, telling him to resist her message as strongly as he can, so that they do not "get" (i.e., convert) him without significant cost.
BARBARA. You're not getting converted, are you?
BILL [with conviction] Not ME. Not likely. Not arf.
BARBARA. That's right, Bill. Hold out against it. Put out your
strength. Don't let's get you cheap.
"But people don't never speak proper anyhow..."
Finally, grammatically speaking, "cheap" should be "cheaply", as it's an adverb for "to get".
However, Shaw is representing speech, and speech is much more fluid when it comes to grammar. This particular use of adjectives for adverbs is common even today in South-Eastern England, to the point that when I read that line, I immediately imagined a woman from the East End of London saying it.