How does table alias names affect performance?

Almost not at all, the performance impact is negligible, but you'll have a much better time reading the query. It's just for your convenience.

The performance impact is allocating a few kb of memory to store alias names, etc. in the SQL Server program itself. Compared to the rest of the operations needed to execute your query, this is almost nothing.

The alias doesn't affect performance in any practical or measurable way at all (italics added on edit). That is, it would add a barely (if it all) measurable delay to query compilation. Once compiled (and re-used), it has no effect.

An alias removes ambiguity when you have more than one table because you know which table it comes from. It also prevents future table changes from breaking a query. Say, you add an audit column to one table where it already exists in another table. Queries without aliases using both tables will break.

An alias is also mandatory in some cases e.g. schema bound views.

The SQL parsing engine (that reads all queries before executing them, and uses the information to cache the compiled queries in the future for faster execution) is the only thing that looks at the aliases, and uses it to help remove ambiguities in symbol lookups. The system would already produce symbols, just like any other compilable statement in any other language, when it's being parsed prior to execution-storage.