Hello, Ken. Welcome back!
I have the impression that almost all English native speakers use the expression not a) but b) when they launch a meeting, presentation, etc.
a) Let's start.
b) Let's get started.
Yes, that's true, though we do sometimes use "Let's start," even in that context. Generally speaking, however, instead of "Let's start" we use "Let's begin."
I have two questions.
1) Is a) a correct sentence?
Yes, sentence (a) is correct.
2) The reason for the use of the verb "get" in b)?
I suspect "get" makes the voice of the sentence passive, so that allows the speaker to omit an objective.
No, "get" is not a passive auxiliary in "Let's get started," which is not even a sentence in the passive voice. Rather, "get" is a copula (linking verb) there, similar in meaning to "become," and "started" has an adjectival meaning. The sentence is saying, in effect, "Let's become started."