The meaning of $$\alpha^{+L}$$ for $\alpha$ an infinite ordinal (countable or not) is just "The cardinal successor of $\alpha$ as seen by $L$." If that isn't clear, you may prefer the following phrasing:
"The unique ordinal $\beta$ such that $L\models$ "$\beta$ is the smallest cardinal greater than $\alpha$"."
There are a few things to note here (and the first addresses what seems to be your specific confusion):
The superscript-+ notation should always be understood (unless otherwise specified) as referring to the cardinal successor. Some texts use it to denote ordinal successor, and embarrassingly it is fairly standard to denote admissible ordinal successor as well, but the default interpretation should always be cardinal successor. The $L$-superscript, meanwhile, is just the usual relativization to $L$; it could be more clearly written as "$(\alpha^+)^L$."
All cardinals are ordinals (precisely, "cardinal" is shorthand for "initial ordinal"), so there is no type error in comparing cardinals and ordinals. That said, if you want to avoid mixing language like this, we could instead define $\alpha^{+L}$ as "The smallest ordinal $\beta$ such that $L\models$ "$\beta>\alpha$ and there is no injection from $\beta$ to $\alpha$"," which is equivalent.
While $L$ and $V$ may disagree about what is and is not a cardinal, they will not disagree about what is and is not an ordinal. Similarly, $L$ and $V$ will agree on ordinal comparison and ordinal arithmetic. This is why it is perfectly fine that my "unique ordinal $\beta$" clause appears outside the scope of the "$L\models$" clause.
There's nothing special about $L$ here; we could redo all of the above with respect to some other inner model $M$ if we liked.