Turning arguments into exponents

Here's an explanation of why you get {4}. As mentioned above, your problem is the use of Rule instead of RuleDelayed, which means the RHS of the rule evaluates before the rule is used. So, let's see what the RHS of the rule evaluates to:

-{a} //FullForm

List[Times[-1,a]]

So, when evaluated, the rule becomes:

L[a___] -> {Times[-1, a]}

Now, it should be clear what happens. When the rule is applied, a gets bound to the sequence -1, -2, -2, and so the output (before evaluation) is:

{Times[-1, -1, -2, -2]}

which evaluates to:

{4}

It's just a matter of using RuleDelayed, or rather :> instead of ->.

L[-1, -2, -2] /. L[a___] :> x^(FromDigits[-{a}])

x^122

L[-1, -2, -2] /. L[a___] :> -{a}

{1, 2, 2}