What is in it? Today is colder than yesterday. What do you mean yesterday?

Both refer to the weather. Of course, the former refers to "today's weather" and the latter refers to "yesterday's weather".

Yesterday can be understood as "yesterday's weather".

Most comparative adverbial clauses after than use ellipsis grammar, and only the subject or subject and the corresponding be verb/auxiliary verb/modal verb are reserved. When you keep the subject, you can even change the pronoun as the subject into the objective case (that is, use than as a preposition).

Another example is that he is taller than me. He is taller than me.

He speaks English better than I do. He speaks English better than I do.

The first sentence uses the be verb is, so the sentence after than also uses the be verb am.

The second example is because the previous sentence uses the behavior verb/entity verb speaks, so the clause after than keeps the auxiliary verb do.