For example, in the sentence “Let me help you” there are two verbs, let and help. Me is an objective pronoun, so that would sound as though it can’t be the subject. But you doesn’t seem to be the subject either because it’s receiving the help. Typically, one finds the subject by asking “Who’s doing the verb (action).” But when you have two verbs, how do you find the subject?
When you see the word "let" at the start of a sentence, it's almost always going to be the imperative. The imperative is like giving an order; the subject of the imperative is never written, but can be inferred from the context, and most commonly is "you".
"Let" means the same as "allow", but whereas "allow" is followed by the infinitive in the form of "to + verb", "let" is followed by the bare infinitive (i.e. without "to"). You can think of the imperative "let" as being shorthand for "you must allow". Note that English is expert at using shorthand; note also that "must" is like "let" in being another of the small group of verbs that is followed by a bare infinitive.
To make it easier to see how all this works, let's reformulate your sentence:
Let me help you = You must allow me to help you.
Now you can see that the first "you" is the subject, "me" is the object of "you must allow" [but also implicitly acts as the subject of "to help"], and the second "you" is the object of "to help".